tag:blogger.com,1999:blog-22104970642073783862017-12-20T04:43:28.837-08:00Teacher Educator BjørnI am a teacher educator in mathematics. In this blog I will write about mathematics, teacher education and topics connected to this.Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.comBlogger99125tag:blogger.com,1999:blog-2210497064207378386.post-54975850393643292882017-02-04T23:33:00.001-08:002017-02-10T03:41:37.268-08:00CERME 10 Day 4The first TWG session on Saturday consisted of four ten-minute presentations, followed by discussions. As I had one of the presentations, it's a bit hard to give details on them (one does get a bit too occupied with one's own presentation in such circumstances). They were:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Rodolfo Fallas-Soto: "Variational strategies on the study of the existence and uniqueness theorem for ordinary differential equations"<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Me: "Design research with history in mathematics education"<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Antonio Oller-Marcén: "Analyzing some algebraic mistakes from a XVI century Spanish text and observing their persistence among present 10th grade students"<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Katalin Gosztonyi: "Understanding didactical conceptions through their history: a comparison of Brousseau's and Varga's experimentations"<br /><br />In the discussion, some of the points were:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Tradition and contextualisation are important - the traditions researchers come from are important (in the case of my design research project). It is important to be clear about the context of them (but on the other hand, it is also important for design research projects to consider and describe which context they may be relevant for).<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• There was a chicken-and-egg-discussion on what comes first in historical research - the question and/or method or the data. (Arguably, all the world is data - or; you can say that they only become data when they can be helpful in answering a question someone poses.)<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• In what way do theoretical frameworks "work"?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• What to do once epistemological obstacles are identified? Should we face or avoid them (until students are "hungry" - why feed them if they're not?).<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Design research - can it be called a "theoretical framework" (as the chairs did in their framing of question for the group discussion). (My answer would be no. A participant also said that it could rather be seen as a framework of aspects to be thought of in such projects.)<br /><br />The next part of the programme was a plenary panel. The panellists were Marianna Bosch (Spain), Tommy Dreyfus (Israel), Caterina Primi (Italy) and Gerry Shiel (Ireland). The topic of the panel was "Solid findings in mathematics education: what are they and what are they good for?" Marianne Bosch was the chair. The background for the panel was EMS' series of articles on "Solid findings in mathematical education". "Solid findings" are defined as important contributions, which are trustworthy and that can be applied. The panel wanted to examine the notion of "solid finding" and consider possible utilities and weaknesses.<br /><br />Tommy Dreyfus pointed out that there are not many review articles in the field of mathematics education. The European Mathematical Society (EMS) decided to help remedy this. (The articles are in the Newsletter of the EMS issues 81-94.)<br /><br />One example: we know that many students "prove" a universal statement by providing examples, across many age levels and countries, including teachers. We call this "empirical proof schemes". But to be called "solid", an explanation is also needed, and here the explanations are varied. But the main criteria for being "solid" holds. Another example: concept image. Students tend to think with their personal image rather than the definition. This occurs at all levels, in many countries, for almost 40 years and across many topics of mathematics. These are often formed by prototypes. Instruction plays a (limited) role. These findings can be considered "solid".<br /><br />Solidity cannot be "proved", expert opinion is crucial, and experts from several fields should be consulted.<br /><br />Caterina Primi talked about how psychometrics could contribute to solid findings in mathematics education. We often measure something else than the trait we are interested in - for instance signs of anxiety, even though it is the unobservable trait anxiety we are interested in. Of course, we can create instruments to try to measure the trait based on them, and these can also be used to find differences between groups. (And so on. It is hard to see how this rather elementary discussion of psychometrics contributes much to the general discussion of solid results - unless her talk is an implicit argument that psychometrics are more important than other research approaches to get solid results - as many would of course say about their own pet approach.)<br /><br />Gerry Shiel's perspective was whether outcomes of international assessments (PISA) can contribute to evidence-based decision-making. Are PISA findings solid? On the one hand, it is huge (more than 500 000 students have contributed to it). He gave an introduction to PISA and how it tries to be an evidence-based series of studies including testings. He gave an example of how Ireland's performance in TIMSS changed over time, with a significant dip in 2009. This dip has not been explained. Ireland rebounded, while other countries had a dip in 2015 when digital testing was done. Also, Ireland has an increase in the gender difference between boys and girls, which is hard to explain. PISA results are used to inform policy - and PISA surprisingly tries to impact teaching directly by publishing their speculations on what can be inferred by the data.<br /><br />In the discussion (which did not work very well, because of a somewhat confusing combination of "questions" from the floor and "questions" sent electronically), it was asked "solid for whom" - implying that what is solid for researchers may not be solid for teachers (and vice versa). This is an interesting point. Gabrielle Keiser mentioned that we need some methodology for writing review papers - it is a very difficult task, and for instance quantitative analyses are not always helpful.<br /><br />(But in hindsight, it is easy to see that this topic invites people to promote their own research or conception of research...)<br /><br />The last part of Saturday (before the gala dinner) was the last session of the TWG. First, there was a part where participants talked about planned or ongoing projects with calls for cooperation. Then we talked about future conferences, where I presented the plans for ESU8 in July, 2018. Plans for the HPM satelite conference to the ICME conference in Shanghai 2020 were presented - it will be somewhere in Asia. Then the process of the proceedings were discussed, and finally there was discussion on the report of the conference, the result of which will of course be seen in the proceedings of the conference.<br /><br />Due to travel arrangements, for me the conference ended with the gala dinner on Saturday evening (which had much Irish music and rather less talk). Thus, this is the place for summarizing the experience. This was my first CERME conference, and I realized that CERME is not really one conference, it is rather ~25 mini-conferences under one roof and with shared amenities and a few common talks. This means that it in one sense is an intimate conference in the same way as smaller conferences are. However, getting the intimate feel demands some consicous choices - not to switch groups no matter how interesting the talks going on elsewhere are, and to try to socialize with people in the group and not be tempted to only socialize with the people you already know. Then, the CERME experience is quite different than for instance ICME, which is a smorgasbord of interesting talks where you risk never running into the same people twice (even though even ICME has some working groups, of course, so I am exaggerating a bit).<br /><br />Dublin was great, the LGBT guided tour was great and the atmosphere throughout was also great. I did learn some new things during the conference, of course, but most importantly, I think, it solidified my determination to try to focus more in the future. I want to spend my research time to get deeper knowledge in some areas rather than having many parallell projects with different foci. I'll see how this works out...Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-51411746380741096882017-02-04T07:37:00.001-08:002017-02-04T07:37:49.432-08:00CERME 10 Day 3Day 3 consisted solely of TWGs and an excursion. The first TWG session was devoted to discussion on the draft chapter on this group for an ERME book. It was introduced by Uffe Jankvist, who has written the chapter with Jan van Maanen. I did not note down anything from that discussion - but I was perplexed to be put in an "old-timers" group despite this being my first CERME. :-) (My feeling of being "young" was destroyed due to my participation in similar conferences since 2000...)<br /><br />The rest of the morning session was spent on participants sharing informaion on important publications that the others should know of. I have notes of this somewhere, but we were also promised an email later summarizing this.<br /><br />The second session started off with Renaud Chorlay's paper, about using parts of Nine Chapters in teacher training. He has three goals for working with this problem (which may be a problem, as students often focus on at most one). Liu Hui gave two justifications for multiplication of fractions, the second of which could probably be used in teaching, in my opinion. The use of a semantic embedding (word problem) is a resource, but also a worry as it can decrease the generality. Renaud argued convincingly that this example can be useful for discussion with teacher students, even though (according to him) perhaps not useful for direct work with children. I am a big fan of Renaud's work and am happy that he is now working in teacher education, as it means that his work - which is as always historically solid - now includes sharp analyses of what might be the use of the historical examples in teacher education.<br /><br />Next, Regina Moeller and Peter Collignon talked on their paper which concerns the work on infinity with children. The concept has a long history, while teacher education students tend to have only the epsilon-delta based concept. (Of course, this is context-dependent - most Norwegian teacher education students would look at you wide-eyed if you mention epsilon or delta.) In their opinion, teachers need to know other conceptions that may be closer to the steps children go through. They look especially at Hilbert and Cantor - including the hotel of Hilbert, of course. The work can make students more aware that there exists different conceptions that they have not learned and to be more open-minded.<br /><br />Then, Rui Candeias presented "Mathematics in the initial pre-service education of primary school teachers in Portugal: analysis of Gabriel Gonçalves' proposal for the concept of unit and its application in solving problems with decimals". This is part of a larger research project comparing different textbooks for teacher training. He presented in detail the steps adviced by Gonçalves. (Which makes me think that it could be a good idea to study historical teacher guides in Norway to point out to students the evolution of the field of mathematics education when it comes to concrete advice given to students.)<br /><br />Maria Sanz gave the last presentation of the day; "Classification and Resolution of the Descriptive Historical Fraction Problems". She proposes a classification of the problems based on which methods can be used to solve them. It is unclear to me what this classification brings to the table - other aspects (known/unknown context, size of numbers, distractors included and so on) could be as important for practical use in classrooms. In the discussion, she was asked about connection to the mathematics education research on the same issues. It was also mentioned that in some countries they are "banned" from textbooks, while in others they are obviously not banned.<br /><br />Some comments that turned up:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• What can these examples bring to teacher training? The common denominator seems to be that they are in a preliminary phase - but they can work to show students that problems are not something to be solved but rather something to be analysed to decide whether and how to use in their teaching.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Could students solve and classify problems in the way of Maria themselves? Would that be more useful than being presented with a classification?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• History can be a good tool to connect algebra without the symbolism with algebra with symbols.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• A book by Brian Clegg on infinity was recommended.<br /><br />I do think that a closer collaboration between maths ed people and history of mathematics people is called for. In some cases, we see discussions on how historical sources can be used in teaching of subjects where there exist a huge amount of literature in the field of mathematics education, but where this work is disregarded. This is every bit as bad as the huge number of papers in mathematics education that completely disregards the history of the subjects that they want to discuss.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span><br />This was the end of the third day. Well, not quite. I was lucky enough to take part on the "lavender walking tour", which was a walking tour of Dublin LGBT History. We saw the Oscar Wilde monument, the Parliament, Dublin Castle, the national library and many other places of importance. We got detailed and enthusiastic information on the liberation fight, including the disgraceful attitudes of the government when activists tried to save lives by distributing condoms (which were illegal at the time). Today, Ireland has moved in a liberal direction and is one of the few countries where gay marriage has been decided in a referendum - although relgious fundamentalists still have a role. The tour ended at a gay pub where we got to continue the discussion over some Irish refreshments.<br /><br />CERME is the second big international mathematics education conference in less than a year with something concerning LGBT issues on or near the programme. I do hope that this is an emerging trend.<br /><div><br /></div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-58480313643844784862017-02-04T07:08:00.003-08:002017-02-04T07:08:50.940-08:00CERME 10 Day 2The second day of CERME 10 started where the first one ended - with a TWG (topic working group session). Please excuse my extremely short descriptions of the papers - the authors were just given ten minutes to remind participants of their papers as a basis for discussion, and I do not have the time to go back to the papers to give more detailed accounts. First, Kathy Clark talked on the very interesting TRIUMPHS project, a big design research project based on original sources. At this time, the project reports on a pilot study in the first year. I notice an inteesting focus on meta-discursive rules and on views of mathematics. They use Törner's aspects and his instrument - but the number of students included in the analysis at this point was small. It will be interesting to follow the project in years to come!<br /><br />Rainer Kaenders talked about "Historical Methods for Drawing Anaglyphs". In this project, students draw 3d drawings using historical methods. The point was not to learn the methods, but to understand the mathematical principles in order to be able to do the drawings. Again, this was an interesting project giving ideas for working on geometry in new ways. Kaenders had used this in extracurricular activities with students, for which it seemed well suited.<br /><br />Thirdly, Rita (Areti) Panaoura talked about the paper "Inquiry-based teaching approach in mathematics by using history of mathematics - a case study". In Cyprus, which has a centralized school system, history of mathematics is seen as a tool to investigate the mathematical concepts. She reiterated Siu's reasons that teachers hesitate in using HM. She gave examples of teachers' attitudes and knowledge. Teachers could not connect the HM and the inquiry-based teaching approach which was also mandated. Understanding what teachers need in order to include history of mathematics in their teaching, is very important in order to implement HM in teaching. As such, I find this paper interesting. A participant questioned whether the use of Egyptian multiplication is helpful. I think that depends on the goal. According to Rita, there are no teacher guide saying what the point is, therefore it is difficult to see if the example is well-chosen or not - and difficult for teachers to use it in a meaningful way. Thus, this paper shows the problem of giving teachers resources without giving them the reasoning behond them.<br /><br />The fourth presentation was of the paper "Teaching kinematics using mathematics history" (Alfredo Martinez). This is a paper concerning a reconstruction of a method of measuring time which Galileo may have used. Students were able to measure time using a rhythm, thereby being able to recreate Galileo's results. It is a bit unclear to me if this really fits in the history of mathematics group or would rather fit in a history of science group (at some unspecified conference), though.<br /><br />Then there was a group discussion and sharing. Some points:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• It is a shame that the scaffolding was not there for the teachers or students in the Egyptian multiplication example to see the connection to our algorithms.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• What "scaffolding" is needed? Notes to teachers and workshops are parts of the project Kathy talked about. Also, use of history of mathematics should also be included in teacher training.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• A historical document is not necessary, historical problems (without giving the actual source) worked on with students are also useful. But what difference does the source make? (Of course, many authors have written extensively on this.)<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Can all topics be taught using history? Are there too big obstacles in some cases?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Can we do good history and good mathematics at the same time? (My answer would be that we are never "perfect" in the classroom, teaching is always full of compromises. So there is a question of what is good enough.)<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• The geographical and cultural distance is important. Is Greek mathematics more motivating for pupils in Greece?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• How much of the original context must a teacher understand?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Choice of examples: should they be "exemplary" or could we have "fringe" examples? Papers that are most interesting from a historical point of view, may not be the best ones from an educational point of view.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• How do teachers come to have materials that they can use? And how do they (learn to) orchestrate the classroom experience?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span><br />Then, there was time for another plenary: Lieven Verschaffel on "Young children's early mathematical competencies: analysis and stimulation". Researchers today believe that children have a "starter kit", object tracking system and approximate number system (ANS). Gradually, there is a development towards a symbolic representation. There are significant correlations between numberical magnitude understanding and early mathematical achievement.<br /><br />The ordinality aspect of number is neglected in the cognitive neuroscientific work. But research suggest stronger correlation/predictability between ordinal aspect and mathematical skill. For instance Hyman Bass argues for developing number based on measurement. Basing the number concept on cardinality means that later developments, such as fractions, will be more difficult.<br /><br />There is also more interest in children's understanding of basic arithmetic concepts and relations. There is little research on the consequences of this for later mathematics learning. Nunes et al (2015) is an exception.<br /><br />Other researchers have looked at pattern and structures. Mulligan et al (2015) is the most comprehensive, looking at children's awareness of mathematical patten and structure (AMPS). A related intervention study shows no improvement in general mathematics achievement.<br /><br />The research studies mentioned so far look at children's abilities, not their dispositions. (I.e. Asking children to look for a pattern, not measuring whether they see the pattern without a prompt.)<br /><br />SFOR (spontaneous focusing on quantitative relations) - individual differences, and has a direct effect on mathematical results at end of elementary school. Several other such FLAs (four letter acronyms) were also mentioned- we do not know much about their development and interrelationship.<br /><br />Then he went on to talk on domain-general (not domain-specific) abilities, such as attention, flexibility, inhibition, working memory etc. There is evidence of these abilities' importance - to a greater degree than domain-specific abilities.<br /><br />Other aspects mentioned in the talk was the role of parents and early caregivers, preschool to elementary school transition, and the professional development of caregivers and teachers. He concluded by listing a whole range of important aspects which need to be further developed in years to come.<br /><br />For the third session of the TWG, the first person was Luciane de Fatima Bertin, presenting the paper "Arithmetical problems in primary school: ideas that circulated in São Paulo/Brazil in the end of the 19th century". She highlighted the notion of appropriation and the notion of purpose. The word "problem" is undefined, but seems to be synonymous with "exercise", so it has no connection to the modern understanding connected to "problem solving". There was no discussion in the journals analysed on the use of problems in teaching.<br /><br />Asger Senbergs<span class="Apple-tab-span" style="white-space: pre;"> </span>talked on his article "Mathematics at the Royal Danish Military Academy of 1830". His article is based on his Master thesis. The research was based on his curiosity about why mathematics became the main topic when Denmark created a military academy. The value of mathematics as a goal in itself was prominent - not just as a tool for action on the field.<br /><br />Ildar Safuanov's paper "The role of genetic approach and history of mathematics in works of Russian mathematics educators (1850-1950)" was up next. The paper details early Russians ideas on the genetic approach. The genetic approach was connected to the idea that pupils should not just witness but also create mathematics, and was included in the guidelines for mathematics teaching after the 1917 revolution.<br /><br />Tanja Hamman talked about ""Sickened by set theory?" - About New Math in German primary schools". The title is from Der Spiegel from March 1974 ("Macht Mengenlehre krank?"). She has looked at textbooks and teacher guides from West Germany to see whether the main ideas were present in the textbooks. Traditional education did influence the implementation, it is not possible to create a clean slate when dealing with teaching.<br /><br />Then, it was time for group discussions. Here are some points from the discussion:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Do we see history of mathematics education mainly as part of general history, part of mathematics education or as part of history of mathematics?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• It is interesting to look at historical cases to investigate conditions for ("successful") implementation of educational reforms. (Which is part of the value of history of mathematics education for teacher education?)<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• How does it matter that a subject has a history? Does it provide a knowledge base to look at your subject?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• Who decides what are popular and unpopular subjects? What are the forces behind which topics are in vogue at a given time?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• When you know more about the past, you have more tools to deal with the present.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• New Math - was it never, anywhere, implemented as intended, with the intended outcomes?<br /><br />Thus ended the second day of CERME. Although most participants probably continued their discussions into the early hours of the next day, I returned to my hotel room to prepare for the university board meeting next week. It is necessary to mention this, as some colleagues have developed an unhealthy interest in my nightlife while in Dublin... :-)<br /><div><br /></div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-16579869059899785262017-02-04T06:41:00.002-08:002017-02-04T06:41:44.714-08:00CERME 10 Day 1CERME 10 was my first CERME, taking place at Croke Park in Dublin. With a capacity of more than 80000, the stadium had plenty of space for the 800 participants. The opening ceremony included short adresses from various dignitaries (of course, including the leaders of the groups actually doing the work of preparing the conference). For instance, we learned how Hamilton got a key insight (concerning quaternions) by the Royal Canal (which passes just outside the stadium). In addition, there was some beautiful Irish music, of course.<br /><br />The first plenary lecturer was Elena Nordi. Her title was "From Advanced Mathematical Thinking to University Mathematics Education: A story of emancipation and enrichment". She opened with an image from the Coen film "A serious man" - pointing out the popular conception of what university mathematics teaching look like: a professor filling a blackboard. University mathematics teaching today is much more varied than that - the demands on the teachers are quite varied. In her talk, she wanted to give an overview of the CERME work on university mathematics since the first CERME, in a way she called "impressionistic" and personal.<br /><br />She pointed out that the field is quite young, for instance important papers such as Yackel & Cobb ("Sociomathematical Norms, Argumentation, and Autonomy in Mathematics") arrived in 1996. She pointed out that research on university mathematics education has in this period been moving away from being a "hobby" done by mathematics professors without a connection to the general mathematics education research. However, she also mentioned how her field differs from other fields in that there is a less clear distinction between teacher and researcher - the university lecturers are also often researchers. However, she did not fully go into the implications of this.<br /><br />Her (rapid) talk discussed a huge number of papers from different CERME conferences, pointing out developments. For me, who is not doing research on or teach advanced mathematics, the talk was so full of unfamiliar names and developments that I will not attempt to summarize here. Sadly, the speed of her talk also excluded some participants - not all of which speak English on a daily basis. (In fact, 50 countries were represented in the conference.)<br /><br />The main feature of the CERMEs are the TWGs (Topic Working Groups), which one is supposed to stay loyal to throughout the conference and which takes up most of the conference time. The first session of the TWG took place at the end of the first day. Renaud Chorlay gave a quick introduction to the working of the group.<br /><br />After we had all introduced ourselves, we were ready for the first paper. That was Elizabeth de Freitas' paper called "A course in the philosophy of mathematics for future high school mathematics teachers". She talked about a course she has given for three years ar Adelphi University in New York, which was actually an alternative to a history of mathematics course. One important aspect is the philosophical paper students have to write - where they have to take a stand and defend a position on one central question from the philosophy of mathematics. Maurice O'Reilly presented his paper on "Multiple perspectives on working with original mathematical sources from the Edward Worth Library, Dublin". He stressed the scaffolding of students' work - helping and encouraging the students reading unfamiliar sources (to them) in foreign languages. These were short presentations as we had all read the papers in advance. Then we started discussing the expected and actual impact of the teaching projects. The discussion centered on whether there are ways of collecting data and convince others of the potential value of such approaches. Here are some points:<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• The researchers had some data that could have been analysed to shed light on the potential. However, as some of the assumed values concerns students' long-term approach to and image of mathematics, maybe longitudinal studies are neccessary?<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• In some cases, The visceral reactions of the students are powerful but not measurable? Some participants in the group recognized their own reaction in students' reaction.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• The role of the teacher seemed to be different here than in "usual" teaching. The projects can give ideas on how to teach to avoid the students' imitation.<br /><span class="Apple-tab-span" style="white-space: pre;"> </span>• There is a pull to prove effectiveness, but also a danger of being drawn into the metrics. We need more research that convinces others than ourselves, but we also need development and ideas that can later be explored more. So papers such as these are valuable although they may not convince others.<br /><br />That was already the end of the first day at CERME. Three more blog posts will follow.<br /><div><br /></div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-6071560791500870932016-07-31T12:15:00.004-07:002016-07-31T12:15:46.450-07:00ICME13 Day 7 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The 7th day of the ICME13 conference was short. First a plenary panel on "transitions in mathematics education". Panellists were Ghislaine Gueudet<span style="text-decoration: underline;">,</span> Marianna Bosch, Andrea diSessa, Oh Nam Kwon, and Lieven Verschaffel. The panel's theme - transitions - has many interpretations, including transitions between themes (arithmetic to algebra), transition to formal proof, transitions between school levels, transitions between contexts, for instance language contexts, transitions between curricula. In this panel, they focused on transitions as conceptual change and on transitions of people as they move between social groups. To look at this, they had epistemological, cognitive and socio-cultural perspectives. The result of the work is a survey: "Transitions in </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Mathematics Education".</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">diSessa started by talking about "Continuity versus Discontinuity in Learning Difficult Concepts". Bachelard talked about epistemological obstacles, while others talks about "pieces and processes" instead, arguing for more continuous change. Misconceptions belong to the left side of this divide. The answer to the discussion may be found in microgenetic perspective (J. Wagner) - some research find incremental learning across many contexts, for instance when trying to learn the law of large numbers. diSessa thinks the continuist side will win, which will mean we will look at resources more than obstacles or misconceptions. (Personally, I'm not sure I believe any side is 100 percent right. Why can't both be right in different contexts? That is, that some development is gradual or stepwise, while other develop is mora abrupt and discontinuous?)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Kwon talked on "Double discontinuity between Secondary School Mathematics and University Mathematics". The double disconuinity concerns first moving from secondary school mathematics to university mathematics and then back (as a teacher). She then discussed Shulman, Ball and then Heinz et al ("School Related Content Knowledge") and Thompson (Mathematical Meaning for Teaching Secondary Mathematics).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">While listening to her talk, I wondered if it could it argued that the Norwegian system, in which teachers for (lower) secondary schools are educated in teacher education programmes in which mathematics and mathematics education courses are merged, and where the mathematics is not "university mathematics" as such, avoids such double discontuinities? In fact, the suggestions she ended her talk with seemed to align almost exactly with the Norwegian model. But it must be stressed that for upper secondary schools, the problem is present also in Norway.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Borsch talked on "Transitions between teaching institutions". Individual trajectories are shaped by the institutions they enter, their activities and settings. The transitions between primary and secondary and between secondary and tertiary education, are much studied. There are many different levels of analysis which are present in the literature. The main differences between primary and secondary are pedagogy (interaction, autonomy, transmissionist) and discipline (specialist teacher, more division between subjects). Many proposals for smoothing the transition are found in the literature (and in the report).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The differences between secondary and tertary education are similar to differences between primary and secondary. In addition, teachers are researchers. There is more research on particular topics (i.e. Algebra) and there are more proposals. "Bridging courses" are discussed, but university mathematics content is rarely questioned. It is not clear whether all perspectives are equally represented in the literature.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Verschaffel talked on "Transitions between in- and out-of-school mathematics". Much learning and use og mathematics take place outside school. While this has previously been "romanticized", currently, more interest is in what happens at the boundaries. Of course, part of this is research on the didactical contract when playing the game of word problems, in which out-of-school experiences are often unwelcome. There are efforts to facilitate and exploit transitions, for instance RME, "funds of knowledge", Greer. Then the panel ended with a discussion, based on question from an internet forum, at which point I stopped making notes.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then, there was the closing ceremony. Secretary general of ICMI, Abraham Arcavi, had a warm and pleasant speech closing the conference. Then there were many speeches of thanks to different contributors. There was a presentation of the next venue for ICME (Shanghai ). And finally, a wonderful musical number.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">What did I get out of ICME13? (And why will I go to ICME14 in Shanghai in 2020?)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">As I noted earlier, there are many possible outcomes from such a conference, and I can summarize some of them, in no particular order of importance:</div><ul style="direction: ltr; margin-bottom: 0in; margin-left: .375in; margin-top: 0in; unicode-bidi: embed;" type="disc"><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Meeting new (or old) people with which future collaborations are possible. At this conference, I can think of at least four people I didn't know before, with whom there can possibly be some sort of collaboration some time in future.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Getting ideas for future research projects: During the conference, I wrote a list of nine research and development projects that I would like to do. Not all are new ideas, but many have been expanded while I've been here. And some are directly adapted from talks here. I only hope I can follow up on some of them when I get back home...<span style="mso-spacerun: yes;"> </span>Hugh Burkhardt's talk on design research was very inspiring, and I want to do something in that direction (more systematically than I've done before).</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Getting ideas for my own teaching (which can of course also turn into research projects): Marjolein Kool's project on making students creating non-routine mathematics problems, Megan Shaugnessy's and others' ideas on simulations of classroom situations (some of these concerning the notion of "noticing").</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Getting an overview of a field which will make it easier for me to read more about it later: For instance, I hope it will be easier for me to face Brousseau now that I've had a tiny introduction and some context.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Listening to lectures and realizing that I do actually know something: it would be impolite to point out which talks contributed to this realization. But perhaps it is a good sign that for every ICME I go to, I think "been there, done that" a bit more frequently. This is not to critizise ICME (too much), of course not everything can be new to everyone.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Socialize with new or old colleagues: of course, the tendency to cluster based on nationalities may be seen as a problem (and meetings like the LGBT get-together may counteract that a little), but it is a reality that many countries have few meeting-places, so that socializing with colleagues from the same country does make some sense. There's been a lot of that here.</span></li></ul><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The main problem with ICME is of course its size - it's complete lack of intimacy. The trick is of course to find a home in a TSG, but still there will be situations when you're all alone and see hundreds of strangers walk past - which is a challenge for smalltalk-challenged people like me. But still, there is every chance that I'll go to ICME again in four years' time. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">But before that, we'll arrange our own conference (ESU8) in Oslo in 2018. That will be fun - and intimate. And I hope I'll go to CERME in February, 2017. </div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-9068604209561253172016-07-30T23:57:00.000-07:002016-07-30T23:57:04.159-07:00ICME13 Day 6 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">On day 6 (Saturday) of ICME13, the plenary lecture was by Deborah Loewenberg Ball, whose work I have referred to a lot in my own articles lately. She talked on "Uncovering the special mathematical practices of teaching". As usual, I was eager to get signs of her definition of "mathematical" (see yesterday's blog). She organised her story in three parts: a journey, getting lost, finding the way. At some point, many years ago, there was a movement from wondering what mathematics teachers need in teaching to what they use. The "getting lost" part was building tools to "measure" teacher knowledge. On the positive side, ways of studying outcomes of teacher education and professional development were developed, but on the negative side we fell back from understanding practice to looking at knowledge. Also, it may have contributed to separating aspects of teaching, for instance equity. So in a way, we got microscope images of many aspects, but not understanding it from the inside. She argued that we have to work on the "mathematical work of teaching" mathematics. (Still not defining "mathematical" or "mathematics", but at least referring to previous talks this week.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The work of teaching is taking responsibility for maximising the quality of the interactions in a classroom in ways that maximize the probability that learners learn. (This fits quite well with Biesta, doesn't it?) Using "work" is to focus on what the teacher DOES, not on the curriculum, what students are doing etc. "Mathematical" listening, speaking, interacting, acting... are part of the work of teaching. A fundamental part of the work is attuning to other people and being oriented to others' ideas and ways of thinking and being.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She then showed us examples from a grade 5 classroom with 29 pupils in a low-income community. Showing us different sequences, she asked us to pinpoint what the teacher was doing. (What I saw: Getting people to come up. Classroom management. Monitoring. Steering discussion (sociocultural norms), respond to question, saying " it doesn't have to be right") She pointed out how the teacher had walked around the classroom, reading nearly 30 student answers, choosing which to look into) Then Ball mentioned concepts from Cohen and Lohan(?) on assigning mathematical competence, including developing competence. The work of the teacher is partly noticing everything that is not the wrong answer. Translation from learning goals to meaningful language for children. The work of teaching is discursively intensive work: talking while monitoring if anyone understands what you're saying and understanding the answers.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I liked the talk very much. As I saw it, it was a step away from "the ball", which has worked to privilege some kinds of knowledge and hide away others, not to mention skills. The criticism of testing for MKT is also welcome. What comes instead, though, is not clear. I'm afraid "the ball" is too simple and pretty to be replaced by anything messier or more filled with doubt and judgements...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The first invited lecture I chose today was Ansie Harding's talk entitled "The role of storytelling in teaching mathematics". Her talk was geared towards tertary education, which is interesting, but I also hoped to learn things for primary education. Of course, storytelling is also an important way of including history of mathematics in primary school (re discussion group at previous ICME), so also from that point of view it should be interesting, as well as for my cooperation with the L1 teacher of my students is next academic year.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Harding teaches 400-500 students at a time at the University of Pretoria. Here, she first gave the story of storytelling, from about 50.000 BC. (Already at that point someone challenged the veracity of the story, which raises the question of the role of truth in storytelling - which is also part of the discussion concerning history of mathematics in teaching; which role does myths have in teaching?) In her story, she included movies, TV and the internet (not including games) - but the examples she used did not include movies - she stressed the value of low-tech storytelling. Instead, her first story example was the story of the length of a year, leap years and so on. (It is a bit difficult to see the difference between her story and normal lecturing, so maybe normal lecturing should be defined as well.) Features of storytelling: are essy to remember, are compelling, are for all ages, embed value, travel, fuel conversation. A story often has a character, ambition, problem, outcome as well as an emotional connection. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The value of storytelling in education is to entertain, inspire and educate. With tertary students, stories can be used as an intro (but is a bit like having the dessert before the main course), as a "by the way", as a commercial break, as a reward (which is how she uses it). You need to get everyone involved and create a "class feeling". </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Ingredients: maths connection, human element, tale (start, flow, end), humour. Her second example was "How mathematics burned down the houses of parliament".</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">With engineering students, she uses the last ten minutes every week (which contains 200 minutes a week) for story telling. She showed a list of examples, much of which the usual history of mathematics "stories". Telling stories takes some effort (you must make it your own), it is a culture that you foster, there are unexpected rewards.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">5-10 students leave before the stories all the time. She asked students for comments, and 30 students responded. Categories: emotional impact, reward, motivation, subject impact, appreciation (more approachable), bigger picture (more to maths than this course).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She ended her talk with two stories, one on De Moivre and one called "The story of one, five, seven" on van Gogh's sunflowers. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">For me, this talk was an inspiration to dare telling stories more in my own teaching, and to put stress on them, not to tell them apologetically and quickly. However, "making it my own" will probably mean making sure they are relevant to the mathematics and often the history of mathematics that we are working on. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After a break, I was back to TSG47, where there were three talks:</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Caroline Lajoie talked on "Learning to act in-the-moment: prospective elementary teachers' roleplaying on numbers". The observation was made in one lesson in teacher education. Her key concept was knowing to act in-the-moment. (See Mason and Spence 1999.)<span style="mso-spacerun: yes;"> </span>in this context students take role of the teacher and 1-3...students, in front of the whole class. Role play involves introduction time, preparation time (everybody prepares all roles), play time and discussion time. She gave examples on how students, even though prepared, need to improvise. One took a risk in a situation where she was not certain of the answer, while another chose to switch to an explaining mode when he saw that he didn't know what would happen. This came up in the discussion, but that was because students were willing to discuss and "criticize" each other.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Pere Ivars on "The role of writing narratives in developing pre-service primary teachers noticing". "Narratives" in this talk refers to stories about classroom interactions. The hypothesis is that writing narratives will help noticing. Students wrote two narratives, with feedback after the first narrative (and some detail of the feedback given were provided here). The development from the first to the second narrative was more evidence of students understanding in more detail (it is important also to notice that students wrote the first narrative based on observation, the other based on what they themselves taught).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lara Dick on "Noticing and deciding the "next steps" for teaching: "a cross-university study with elementary pre-service teachers." Noticing is a skill teachers need and that they need to develop. Much work have been done with using video to work on noticing, but little on use of student work (such as the previous one). The current project was a three-hour lesson on multiplication. The students saw student examples and analyzed one each and made posters. The students were asked to attend to the mathematics, interpret what they see and make an instructional decision - based on looking at all the posters. Four major themes: gravitation towards traditional teaching ideas, vague decisions, desire for written number sentences, focus on strategy progression. (This seems like a lesson it would be useful to try to copy with my students as well.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(About TSG proceedings: there is a format suggestion from the ICME , but it is unclear how much flexibility there is. The suggestion was few but extended papers, while the TSG organizers would prefer shorter but more papers.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">These talks remind me that key concepts to meet for new teacher students are noticing and building on what students know instead of figuring out what they don't know. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After an extended break, I attended, of course, the HPM meeting. Luis Radford gave an introduction to the HPM<span style="mso-spacerun: yes;"> </span>group, its research and publications, while Fulvia Furinghetti presented the history of the group. Then, Kathy Clark, the new chair of the HPM, gave a presentations with her own background and some views on the HPM. She mentioned two projects: ÜberPro project (Germany; Übergangsproblematik) and TRIUMPHS (US; transforming instruction in undergraduate mathematics primary historical sources).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(The proceedings of the 2016 HPM conference are online here: <a href="http://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/History_and_Pedagogy_of_Mathematics_Proceedings_of_2016_ICME_Satellite_Meeting.pdf">http://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/History_and_Pedagogy_of_Mathematics_Proceedings_of_2016_ICME_Satellite_Meeting.pdf</a>)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This concluded the last whole day of the conference. However, discussions on mathematics education and other interesting topics continued for many more hours...</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-70906995924025091462016-07-29T22:09:00.004-07:002016-07-29T22:09:53.091-07:00ICME13 Day 5 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In case you are desperately looking for the Day 4 post, please note that Thursday was excursion day. I chose to make this a Excursion-alone day, as there was a Manet exhibition in the Kunsthalle (right next to my hotel) that I did not want to miss. (It was great - and as so often, a good guide, in this case a multimedia guide, made all the difference. I also don't have much of a memory, so I was surprised at some of the cool paintings in the permanent exhibition, even though I saw them three years ago.) Moreover, of course I had to prepare for my talk on Friday, prepare for my workshops next weekend (at LAMIS in Ålesund) and also generally wake up after Wednesday night. So I spent some time with a cup of coffee and my paperwork in Lange Reihe, but also had dinner with colleagues in the evening.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Of course, the excursion day is also a day when you reflect on the conference so far. The simple question "Was it a good conference?" can be unpacked into subquestions such as Did you meet new (or old) colleagues that you may collaborate with in the future? Did you get ideas for new projects or for your teaching? Did you get an overview of topics that will make it easier to study them further when you get home? Did you get that warm feeling of hearing a talk and realizing that you actually know that topic quite well already? On at least those counts, this conference has so far been a success.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Friday started with a plenary lecture by Berinderjeet Kaur titled "Mathematics classroom studies - multiple windows and perspectives". The TIMSS Video Studies (1995 and 1999) was an inspiration for further classroom studies. They found that teaching varied a lot between different<span style="mso-spacerun: yes;"> </span>cultures - and we are not aware of the differences. The 1995 study showed different patterns for different cultures. The 1999 study introduced a new terminology of wide-angle lense perspective and close-up lense perspective, and argued that both are needed. (By the way, I was lucky enough to get access to data from this study to do a study of history of mathematics in the seven countries. But that is many years ago now.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In the late 1990s, there were a couple of studies involving Singapore, the Kassel project and a study of Grade 5 mathematical lessons. In the Kassel study, there were lesson observations. There was a need of a shared vocabulary when talking about classroom activities (for instance, she mentioned that the term "Singapore maths" does not make sense to her). The study gave a wide-angle view of Singapore<span style="mso-spacerun: yes;"> </span>(pretty much in line with what one could expect). The next study consisted of just five lessons, also without many surprising results.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In 2004, Singapore joined the Learner's Perspective Study (LPS). This study investigated from the perspective of students. Three teachers participated in Singapore, a ten-lesson sequence for each were recorded. To code lessons, they wanted to take into account the instructional objectives of the teacher (which seems to make sense), but a teacher always have more than one objective to a lesson (which many school administrators don't seem to understand). This made for complicated analysis. The close-up lens gave a different picture than the wide-angle lense. Lessons were well structured, objectives were clear, examples carefully selected, student work carefully selected for classroom discussion. In addition, students were interviewed after the lessons. This resulted in a long list of characteristics that the students valued (which seems quite similar to what teachers try to achieve).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She ended by arguing that the stereotypes about Singaporean mathematics are not correct, especially when you use a close-up lense. (This is not fully convincing to me, at least not if based mainly on the study of three highly qualified teachers.) She also noted 2010 survey data that showed that teaching for understanding is strong in Singapore, and the analysis shows clear links between the different instructional practices.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then I attended Michael Fried's talk on "History of mathematics, mathematics education, and the liberal arts". I have heard Michael several times before, and knew I couldn't go wrong there. He started by pointing out that he will not talk about history of mathematics as a tool (referring to Jankvist). Here, the point is history of mathematics as a object of study, not as something to use.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">First, he started with D. E. Smith, who was interested in history of mathematics and also set in motion the idea of ICMEs. In another writing, Smith claimed that the motives for teaching arithmetic is either for its utility or for its culture. History of mathematics should be at the very heart of culture. He used both the "parallelism argument" and history mathematics as a filter showing us the importance of each part of mathematics. Fried argued that Smith's view of mathematics is ahistorical - he looks at mathematics as a set of eternal truths. Hence we see that the view of mathematics is not the result of not knowing enough history. This view of mathematics is what allows it to be a tool. And oppositely: allowing history to be a tool, makes it non-historical.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">He then looked at "religio historici", where original texts are at the heart of doing history. It matters how you consider the past (of course, we know that parts of the past have been deemed as uninteresting by historians at times). The past seen as just what lead to the present (a Whig interpretation of history) is uninteresting, "practical history". Oakeshott and Butterfield wanted us to see the otherness of the past.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This unhistorical history is particularly tempting in history of mathematics, because mathematics is seen as eternal. He stressed the different concerns of historians of mathematicians and teachers of mathematics. The demands on teachers means that teachers will not put modern mathematics aside to teach history. Thus, teachers needs to economize and make history of mathematics useful - thereby almost neccessarily embrace a Whig view.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In 2001 Fried claimed there was a clear choice between teaching Whig history or to drop the idea of being useful. This damns all hope of including history of mathematics in mathematics education.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">However, in principle, one can ask how mathematics education can be conceived to include history of mathematics as an integral part. Here he named people like Radford, Barbin, Jahnke, Jankvist, Clark, Guillemette as examples of people who can be seen to be working on this. (This means fighting simplistic uses of the word "mathematics" as learning methods, which can be glimpsed in some research articles. I have often questioned what people mean by "mathematical" in the phrase "mathematical knowledge for teaching". In my opinion, the history of mathematics is an integral part of mathematics, in the same way that you can't separate literatur from its history in a meaningful way.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then he went on to speak of the history of the "liberal arts", where mathematics was a self-evident part. (He spent a significant time detailing this, including pictures, which I cannot describe here.) The liberal arts was supposed to be what you needed to become fully human. History was never considered part of them. Today, history is considered part of the liberal arts, while mathematics is missing. This could be fixed, and history of mathematics can be seen as a way of solving it. Mathematics education then also becomes a way of reflecting on ourselves.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(In Norwegian teacher education, I'm not sure the argument is exactly the same. We talk about so-called theoretical subjects and so-called "practical and aesthetical subjects". But similar arguments can be made - of course, we know that mathematics is also practical and aesthetical, as history of mathematics can also show.) </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This was a thought-provoking talk. I believe that I will - as usual - end up in a pragmatic view where including history of mathematics in different ways and with different goals will still be better than doing nothing. I believe in teaching history of mathematics as a goal, but that this can still be "useful", and that teaching history of mathematics as a tool can still instill a sense of "real" history. Teaching is never perfect, it is always in need of judgment and a balancing act between different concerns, so why should this area be different?</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then it was back to the TSG. Derya Çelik talked on "Preservice mathematics teachers' gains for teaching diverse students". As she could not come, the talk was sent as a video. She talked about a project with 11 researchers. They analysed PSTs opinions on how often the program provided opportunities to learn about teaching students with diverse needs. 1386 PSTs took part. "Teaching for Diversity" scale was used. In general, the results showed few opportunities, although there were some (significant) differences between regions, with more developed regions scoring higher. This fits with TEDS-M results (also for Norway).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then I presented the work of Eriksen, Solomon, Rodal, Bjerke and me, with the title ""The day will come when I will think this is fun" - first-year pre-service teachers' reflections on becoming mathematics teachers". I did not make any notes during this talk, as everything felt quite familiar... :-) The people present seemed interested, though.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Oğuzhan Doğan talked on "Learning and teaching with teacher candidates: an action research for modeling and building faculty school cooperation". He started by stressing the importance in teacher education in improving teaching practices. High quality field experiences can contribute, while poor quality field experiences will support imitation. The aim in this project is to find ways of improving. They planned a hybrid course where teacher educators and PSTs plan mathematics tasks and activities and apply them in a real elementary classroom.<span style="mso-spacerun: yes;"> </span>(He gave some details on the design of the course which I can't repeat here.) At first, they applied tasks that the teacher educators had made, but then they applied tasks that they had made together. They saw that teacher candidates left the idea that the answers were the most important. They also went from exercises towards discovery, and saw the important role of manipulatives. The teacher involved also used the tasks given in other concepts. For the future, they plan to do something like this in the compulsory course, even if this course does not go on.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Finally, Wenjuan Li talked on "Understanding the work of mathematics teacher educators: a knowledge of practice perspective". She pointed out that there has been a lot of research on PSTs and teachers, but not on what MTEs (mathematics teacher educators) need to know. She used a distinction between knowledge for/in/of practice. Of course, we need to include both mathematics and mathematics education knowledge. The study further hopes that studying what MTEs do, will inform us on what they need to know. (This is a bit doubtful, and moreover, just as with mathematical knowledge for teaching, what the particular MTEs don't know, will be missed in the model even if useful.) They included only six MTEs in this project. All had school experiencem but they had limited knowledge of research. (Which means that the results will probably be different from results in a similar study in another context, with MTEs with another background. And, remember, parts of the rationale of having teacher education in universities, is that the MTEs are actively doing research and development work, so that they will have very different specialities which (hopefully) contributes to their teaching in different ways. Of course, the same criticism can be (and probably has) been directed to Ball's work etc. But even so, the results could be interesting to build upon with data from other contexts.) </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After lunch, I heard Ronald Keijzer's talk on "Low performers in mathematics in primary teacher education". In the Netherlands, there is a third year national mathematics test (because of PISA and TIMSS results, comparable to in Norway). The project investigated characteristics of the students who did not pass this test. (Could it be seen already in the first year?) Both interviews (n=12) and a questionnaire (n=265) were used. Previous mathematics scores predicted score in third year test. Low achievers specialize in teaching 4-8 year olds (not 8-12). Many similarities with other students, but they are less enthusiastic than those who passed the tests.<span style="mso-spacerun: yes;"> </span>They disagreed that they were low achievers or did not work enough or that they didn't get enough support. Rather, they blamed the test (interface, content, preparation, feedback) and personal factors (dyscalculia, stress,<span style="mso-spacerun: yes;"> </span>concentration). So the conclusion is that low performers are often that for a long time, they often blame the test or personal factors. There was a lack of self-reflection. Comment from the room: mindset treatment as a way of increasing self-reflection.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After this, I needed to meet "my old group", TSG25 (on History of mathematics), where I heard several talks throughout the evening:</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">ChunYan Qi talked on "Research on the problem posing of the HPM". She referred to MKT, in particular SCK, and she focused on problem posing based on HM. The research look at 68 problems based on (one?) problem from Chinese history. She listed three strategies for making problems based on history (but I didn't - in the very short time - quite understand why these three were chosen or how we would know that the results would be problems). Then she discussed which strategies were preferred by students and other findings that turned up when analysing their problems.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Tanja Hamann gave a presentation on "A curriculum for history of mathematics in pre-service teacher education". She started by referring to Jankvist, noting that using history as a goal is perhaps a bit more important for teacher education than at other levels, while history as a tool is more a general concern of all levels. But of course, you often work on both at the same time. She argues that HM is important for teachers to influence beliefs, give background knowledge for teaching, building up diagnostic competences, providing an overview of mathematics, being able to identify fundamental ideas, building language competencies.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">They try to implement history in a long-term curriculum, including history throughout the topics as small parts. They also have a special lecture on mathematics in history and daily life. Finally, they use history in exercises and tasks. For all of these three components she conjectured which goals they could contribute to.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">And after a break: </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Jiachen Zou spoke on "The model of teachers' professional development on integrating the history of mathematics into teaching in Shanghai". He presented a model of how teachers, researchers and designers can work together to design teaching integrating history of mathematics. The model was in three dimensions with many colors and many hermeneutical circles, and can not be described in words. It was unclear to me how the model was developed (based on what data), maybe that would have given me better understanding of the model. However, as always, time is limited. (He was also asked how the ("somewhat formalistic") model were developed, but it was not clear, except that the model had been used to describe three different teachers' paths.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">ZhongYu Shen on "Teaching of application of congruent triangles from the perspective of HPM". His project concerns 7th grade in Shanghai, with two teachers involved. The stories of Thales (finding the distance of a ship at sea from the coast) and Napoleon (a similar method) was used as a starting point, in addition to a Chinese measuring unit for length. They designed a lesson based on this, and had a simple survey asking whether they understood and liked this, and not as many liked it as understood it.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Fabián Wilfrido Romero Fonseca on "The socioepistemologic approach to the didactic phenomenon: an example". The example was three moments from history of Fourier series: the problem of the vibrating string, Fourier's work on heat propagation and development of engineering as a science. He showed how activities were based on the analysis of the historical background, where Geogebra were used to investigate. However, the activities have not actually been used, just developed as part of a master thesis.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thomas Krohn's talk was "Authentic & historic astronomical data meet new media in mathematics education". The students in question were in 10th, 11th and 12th grade. An authentic problem was to describe the movement of comets, and this can be used by giving students some historical/astronomical background, as well as the neccessary tools. The astronomical data are often presented in lists in ancient books which are on the internet. As in ancient times, they first made a projections from 3d to 2d, and then they tried to find a reasonable function (often preferring to investigate instead of leaving it to a computer to find.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Slim Mrabet, whose title was supposed to be "The development of Thales' Theorem throughout history", did not turn up.</div><div lang="nb" style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thus ended the Formal parts of day five. </div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-74511076033674361942016-07-27T23:37:00.001-07:002016-07-27T23:37:17.648-07:00LGBT get-together at ICME13 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">For most LGBT (lesbian, gay, bisexual, transgender) people born in the 20th century, "coming out" never ends. Of course, the first few people you tell may be the most difficult ones, but years later, meeting new people still involves small questions in the back of the mind: Should I be open? Will they mind? Should I use gender-neutral words ("my partner") to avoid having to face potential negative attitudes?</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I still feel that way when meeting new people back home in "safe, liberal, politically-correct" Norway, but I feel so even more in international contexts, where people from the whole world sits around the same table. At international conferences, there is the added consideration: I'll only meet the people for a few days every few years, so why ruffle any feathers? Why risk causing someone to be upset? Couldn't we just stay away from "controversial" subjects such as the gender of my husband?</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The concrete result of this is that I (and many others, I assume), feel less free to be ourselves, to mention my husband where it's otherwise natural in conversations and so on. I get (even) more reserved than I usually are. As a result, others get to know me a little bit less, and I get to know others a little bit less. (Anecdotal evidence: there are people who I've met and talked to at conferences over a period of ten years before we realized both were gay.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This long preample is my answer to the question I asked myself when invited to a LGBT get-together at ICME. I thought "Cool. But why?" I thought it would be strange meeting people here based on sexual orientation or gender identity, and wondered what we could possibly have in common. Now I've realized that having such a meeting is just as silly an idea as having "gay pride" events: it is silly, but still neccessary, because not having such meetings means going on with the status quo, where everyone makes their own micro-decisions of not telling anyone so as not to cause (possible/imagined) offence.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The LGBT get-together was yesterday, and it was great. Surprisingly (?) I even met people interested in the same part of maths ed as myself. Many thanks to Pauline and Nils for getting the idea and making the idea reality, and to the HIV prevention center for the venue and help. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I hope I will see lots of the people who were there again, and that this event will contribute to a more inclusive ICME. I also hope these things will be repeated at future ICMEs, as long as neccessary. Silly or not.</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-66864158228672406022016-07-27T08:57:00.001-07:002016-07-27T08:57:17.562-07:00ICME13 Day 3 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Day 3 of ICME13 started off with Günter M. Ziegler's talk ""What is mathematics" - And why we would ask, where one should experience or learn that, and who can teach it". According to the twitterati, it was good and entertaining. I missed it.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In conferences such as this, it is important to break away from the programme to have time to reflect and to interact with other participants. (Unike some other conferences I go to, this conference does not have many workshops or discussion arenas.) Therefore, I skipped the plenary and an additional lecture, but was back on track after that, for four more TSG-talks: </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Skip Fennell talked on "Preparing elementary school teachers of mathematics: a continuing challenge". He started by giving a brief historical background. He than pointed out that in many countries, elementary teacher education leads to teaching one or two subjects, while in the US (and Norway until now), teachers become generalists. Also, in many countries there are a limited numbers of programs, sometimes hard to get into, while in the US there are more than 1700 programmes, some online. There are voices arguing for having math specialists in elementary school. (In Norway, we are switching to a programme where you have to have a master's degree in one subject, but still will teach two or three other subjects.) However, in 20 states, there are math specialists, often called "maths coaches". These positions are, however, vulnerable in touch (economic) times. In the Q&A, he conjectured that there must be significant challenges in having teacher education programmes which provides everything except practice periods online. We all know the difficult negotiations between campus and practice, and these cannot be easier in such a context.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Marjolein Kools's "Designing non-routine mathematical problems as a challenge for high-performing prospective teachers". She started by giving a non-routine mathematical problem. In Netherlands, there is a national test in the third year of teacher education, and this project concerned creating non-routine problems for PSTs to work on. The got PSTs to help them, but it was very challenging for them to produce task of the right level of difficulty. Therefore, the project turned into a project trying to support the PSTs in creating, and this became a design research project. At the end, 4 of 8 students could design non-routine problems independently. Key to this result was master-classes where Ronald created problems while speaking aloud, trying them out and so on. (Although the starting point of this project was a national test, it seems worthwhile to work on this even in countries which are lucky enough not to have such a national test. It would be cool to do something like this in Norway as well...) </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Eda Vula's "Preservice teachers' procedural and conceptual understanding of fractions".<span style="mso-spacerun: yes;"> </span>She referred to Ma, Shulman, Ball etc. as starting points for her talk. The research questions concerned representations and the relationship with and between procedural and conceptual understanding. 58 PSTs participated by doing a 20 item fractions knowledge test. The test obviously unearthed problems, for instance in the meaning of unit. PSTs had good procedural knowledge, but rarely good conceptual knowledge in addition.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Megan Shaughnessy's "Appraising the skills for eliciting student thinking that preservice teachers bring to teacher education". Of course teachers need to be able to find out what their students are thinking. This project takes this point into teacher education - how do we find out what our students think. The idea here is to use standardized simulations. PSTs get a student response and prepare an interaction with this "student",<span style="mso-spacerun: yes;"> </span>then they interact with his student (who is a teacher educator trained and using a response guideline). She showed a video to illustrate how this worked. All PSTs did this, and the videos were analysed, and the findings were really interesting (but I can't summarize them here). The study shows some of the moves that PSTs need to learn, what can be built upon and what requires unlearning. (And of course the videos themselves may be useful to work on thIs.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After lunch, I attended the "thematic afternoon" on European didactic traditions. First, there was a "panel"<span style="mso-spacerun: yes;"> </span>with Alessandra Mariotti, Michéle Artigue, Marja Van den Heuvel-Panhuizen and Rudolf Strãsser. They had identified five characteristics of the the European traditions: strong connection to mathamatics and mathematicians, strong role of theory, role of design activities, key role of empirical research.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Mariotti talked on the role of mathematics and mathematicians. Historically, mathematicians were important figures in discussions on how to prepare mathematics teachers. In the beginning of the 20th century, Klein had a central role (Erlangen Program). Also, after WW2, mathematicians had key roles in the New Math reform(s). Later, mathematicians also played important roles in the French IREMs, for instance, where mathematicians, teachers and didacticians mixed. In Germany and Netherlands, on the other side, mathematicians played a smaller role in turning mathematics education into a research field. The case of proof as a theme of research have been a feature, and this may be influenced by the role of mathematicians.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Artigue talked on the role of theory. She started with the French tradition, which she claimed had a particular importsnce in building a theoretical foundation. These had a systemic vision of the field, a strong epistemological sensitivity, theories mainly conceived as tools for understanding. New development are connected to previous theories. In the Dutch tradition, Realistic Mathematics Education is the main contribution. This is a theory oriented towards educational design. The theories are still refined. In Italy there is s long tradition of action research and collaboration between mathematicians and teachers, now continued in Research for innovation. In the Germsn tradition, the theoretical landscspe is not so easy to synthesize. (The rest of Europe is left out in this survey.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Van den Heuvel-Panheuzen talked about the role of design activities. The French tradition: essential component of research work and based on theoretical frameworks. In the German tradition, design activities took place in context of Stoffdidaktik, evaluation was often left out. Empirical turn from 1970s. Italy: empirical analysis as base for didactical innovations, pragmatic approach. From 1980s innovations have turned to research. In the Dutch tradition, the design aspect is the most significant characteristic. Theory development resulting from design activities, later theory was used as a guide for futher theory development.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Strässer talked on the role of empirical research. Different kinds of empirical research have developed because the complexity of the field. Examples: the large-scale study COACTIV, the case study MITHALAL. Of course, there are also a distinction between qualitative and quantitative methods. The purposes can be prescriptive or descriptive, and they may be developing vs illustrating theory. Action resesearch have been stronger in I and F, while fundamental research, strongest in F, G, N. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In addition to the content of the talk, what will surely be remembered about this evening is the heat and humidity of the auditorium, not least when it was decided to close all windows to close the curtain to improve the visibility of the screen slightly. It is a testament to the participants' dedication to the disipline that so many chose to come back to another session after the coffee break.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I know far too little about the French tradition, so I chose to go there. I'm also struggling to learn French, and hope that I will be able to read some of the French work later. Brousseau, Chevallard and Vergnaud were the stars of this bit of the program. IREMs were established from 1968, leading to a fruitful collaboration between mathematicians and teachers. From 1975: first doctoral programs, from 1980: journals.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The first theoretical pillar was the theory of didactical situations (Brousseau) - with the "revolutionary" idea that the pbject of research was the situation. Centre of observation for research created in 1972. The core concepts had been firmly established through the 1980s. The second pillar: the theory of didactic transposition and the ATD. (Chevellard) presented in 1980, followed by book in 1985. We "embed an activity into a whole whose ecology is studied". The third pillar was probably the theory of conceptual fields (Vergnaud), which seemed to be taken for granted in this meeting.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">There was a talk on research on axial symmetry in France, to illustrate a development. There has been a development from the study of students' conceptions, combining the theory of conceptual fields and the theory of didactical situations to the study of teachers' practices and of their cognitive effects. Now, French research on symmetry is more occupied by the question of language. There was another talk on French research on school algebra. The first studies were in the 1980s, by Chevallard, in terms of didactic transpositions. Three questions: </div><ol style="direction: ltr; font-family: Calibri; font-size: 11.0pt; font-style: normal; font-weight: normal; margin-bottom: 0in; margin-left: .375in; margin-top: 0in; unicode-bidi: embed;" type="1"><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;" value="1"><span style="font-family: Calibri; font-family: Calibri; font-size: 11.0pt; font-size: 11.0pt; font-style: normal; font-weight: normal;">what is school algebra, and what is (or have been) considered as algebra? (Didactic transpositions, institutional constraints)</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">What algebra could be taught? Under what conditions? How implement them? (Reference epistemological models (REM), didactic engineering (new ecologies))</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">How do ICT modify the nature and the way of using, teaching and learning algebra? (Concept of ICT as part of the adidactic milieu, taking into account the instrumental genesis, didactic and computer transpositions)</span></li></ol><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">At this point, the thing turned into a Q&A session in which the questions were unintelligible while the answers were moderately more intelligible. I chose this point of time to leave the auditorium. However, there was one interesting question that I was able to understand. It was something like this: "It seems that the research questions the French pose about algebra are quite similar to research questions elsewhere. In what ways do the answers differ?" I did not catch the answers, but I made up my own: the problem/challenge with having different theoretical perspectives and traditions are exactly that we may pose the same sorts of questions but look at them from different points of view, and we therefore have an inkling that the answers will be (slightly or significantly) different. However, since the answers are themselves given within different theoretical traditions, pinpointing exactly how they differ demands exactly the same comparison and analysis of perspectives (or perhaps even the creation of a "meta-theoretical perspective") that is some of the point of this thematic afternoon and similar work. If we could easily explain what the different traditions accomplishes, we would be at another point of the development than we are. Perhaps.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thus ended the third day of the ICME13 conference. I had a very good experience in the TSG, getting inspiration for further work. The "thematic afternoon" was an interesting experiment, but the different backgrounds of the participants must have made it almost impossible to plan.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After this, there was social gathering again, followed by an (unofficial) LGBT get-together of ICME participants. I chose to post this before these events, though, to avoid the temptation to blog about them</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-388137207618983552016-07-26T23:43:00.004-07:002016-07-26T23:43:53.290-07:00ICME13 Day 2 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The second day of ICME13 started with a plenary lecture by Bill Barton, with the title "Mathematics, education and culture: A contemporary moral imperative". He started by remembering his first ICME (Adelaide) where Ubi d'Ambrosio was the very first speaker he heard. This inspired him. His other starting point was that pleasure is an important part of mathematics - the pleasure of viewing fractals or of seeing a beautiful proof. (Hm - I often think more of the fun of mathematics than of the pleasure. Maybe there is more to be done in categorizing the pleasures of mathematics?) He asked us to look at the Klein project - and to contribute with translations.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">He went back to Ubi's 1984 talk on ethnomathematics. Later, Ubi has spoken about mathematics as the dorsal spine of modern civilization. Our goals should we responsible creativity and ethical citizenship. But how? According to Ubi, we need to include the cultural roots.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Barton invoked Ecological Systems Theory (Bronfenbrenner). He used five levels: microsystem institutions directly involved), mesosystem (links between microsystems), exosystem (wider social setting - for instance financial world...), macrosystem (cultural context etc), chronosystem (pattern of environmental events and transitions, as well as socio-historical circumstances). He also invoked Ecological Humanity - seeking to bridge the divide between science and the humanities. The links between organisms define how the whole system works. He also added Deep Ecology (Arne Næss) into the mix.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><span lang="nb-NO">Barton then distilled this into three principles: perspect</span><span lang="fr-FR">ive, ref</span><span lang="nb-NO">lexive, pleasure. The perspective principle: be aware of other ways of understanding. The reflexive principle: do unto others as you would have them do unto you. The pleasure principle: act so as to increase pleasure. (All of these on both individual and system level.)</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><span lang="nb-NO">Was the mathematical community responsible for the global financial crisis? It can be argued that it was triggered by the models. </span><span lang="en-GB">Do mathematicians love maths edu</span><span lang="nb-NO">cation as much as maths educators are expected to love mathematics? And do we as maths educators nurture our love for the subject? Do school teachers have the time and resources to nurture their love for mathematics? </span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">We all need to make efforts to ensure mathematics education benefits from multiple points of view. ICMEs should be more language friendly. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">He critizised streaming, which research shows increases the effect of socio-economic factors (and so on). He compared this to having a (public) cooking test at a certain age, and the outcome of the test decides who you can eat with and what you can eat for the rest of your life. Fo 14 years we ask students to practice basics without being offered the opportunity to experience the pleasure of mathematics. And we know the effects of high stakes testing. (Of course, Barton speaks in headlines, nuances are not included. But I guess that is okay from time to time, too often we speak with all nuances and no headlines.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After a coffee break, there were lots of parallell invited lectures. I decided on Anthony A. Essien's on "Preparing pre-service mathematics teachers to teach in multilingual classrooms: a community of practice perspective". This is obviously very relevant to my context in Oslo, Norway, and too little of what I know about this is situated in mathematics learning. (Moreover, I do like Wenger's perspectives.) Essien works at Wits University in Johannesburg.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">There is a triple challenge in multilingual maths classrooms: paying attention to mathematics, paying attention to English (or other language of instruction), paying attention to the mathematical language. (I would perhaps add attention to the other languages the pupils know to the extent possible.) How do we prepare PSTs (pre-service teachers) for this?</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Essien uses Dowling and Brown (2010)'s idea of organisational language, relating to empirical field and theoretical field. Wenger (1998)'s communities of practice can be looked at through three dimensions mutual enterprise, joint enterprise, shared repertoire. But Wenger did not develop his theories for teacher education in mathematics. COP theory does not have a coherent theory of language in use. Essien uses Mortimer and Scott (2003) for this: meaning making as a dialogic process (DP).<span style="mso-spacerun: yes;"> </span>He developed cathegories based on literature and data (not possible to summarize here...)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">There are interacting identities at play: becoming a teacher of mathematics/in multilingual classrooms, becoming learners of maths/of maths practices, becoming proficient English users for the purpose of teaching/learning mathematics. Code-switching is part of this (maths practices). He studied privileged practices-in-use. He showed how different universities teach in much the same way, but still encourage development of different identities. (For instance: when defining, one does it purely mathematically, the other focusing on how it could be explained to children. Thus, the second may be expected to support the development of an identity as a mathematics teacher.) The perspective of multilingualism was lost somewhere on the way, however.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">A question from the room concerned whether the teacher educators were explicit in what role the PSTs should assume (as in "consider this as a teacher" or "consider this as a mathematics learner") The person asking suggested that PSTs may easily be confused if this is not clear and explicit. I think this is one of the most important points for getting students to move from their role of pupil into their role as prospective teacher. (By the way, it was really refreshing that there was time for questions, after a day of conferencing with little interaction. The discussion was interesting and fun.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(An idea I got for an early task with my new students this autumn: "What is the most difficult thing you know well in mathematics? How would you explain this to a learner who don't know this too well in advance?" (And as an added perspective "who knows another language better than Norwegian?"))</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(Based on yesterday's talk on design research, I also get inspired to look at the design on history of mathematics-based mathematics lessons, by way of analysis of published lessons or by interviews with lesson designers.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Much of the ICME programme is filled with the TSGs (Topic Study Groups). I usually attend the TSG on history of mathematics, but this time I attended TSG47 (Pre-service mathematics education), because this is where the project I will present on Friday fits in. For a conference this site, it is important that people have a "home" where they meet the same people more than once, but still I'm a bit uncertain about the weight the TSGs have got in this conference - for instance, most of Tuesday was filled with TSG sessions - mostly filled with ridicuously short presentations (10 minutes each). We'll see how it works. (Of course, the weight of the TSGs is a function of the number of the contributions they receive and accept ("my" TSG has 66 accepted contributions), which again is a function of how broad their themes are. So it is not an easy problem to solve.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In this first session, there were four talks. Fou-Lai Lin (with Hui-Lai Lin) presented a talk on "Using mathematics-pedagogy to facilitate professional growth of elementary pre-service teachers". Hui-Lai started by discussing a game where the point is to arrange coins as a rectangular form. (I didn't get the details of the game, but I see ways of doing this so that lots of concepts will be involved, for instance where children get points based on how high the lowest factor can be.) Fou-Lai started by telling that Taiwanese mathematics teachers are not good in mathematics, sometimes even fearful of it. She gave a list of principles for designing PST-courses, which I didn't manage to note down. She showed an example with binary numbers. PSTs should see analogy to elementary student learning. She also stressed the importance of not only that students should read textbooks themselves, but analyse them and conjecture how pupils will struggle with the textbooks.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Secondly, Roland Pilous talked about "Investigating the relationship between prospective elementary teachers' math-specific knowledge domains." The focus of the study was relationships between the domains (and in talking about domains, it seems he was inspired by Ball et all's domains). The project was based on six students and three lecturers. Four domains were identified: curricular knowledge, content knowledge, teaching-related knowledge and knowledge of student cognition. (The problem of such a tiny number of informants is of course that whole domains of knowledge may be missing. If, for instance, history of mathematics is not mentioned by the twelve informants, that may be lost to the model.) He very quickly mentioned some of the relationships, but of course time was too short to get an understanding of this.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thirdly, Jane-Jane Lo had a talk with the title "A self-study of integrating computer technology in a geometry course for prospective elementary teachers". This is a study of her own teaching, analysing the lesson plans and reflecting on her choices in hindsight. You can use tools for technical activity or as conceptual activity. Dick and Bos discuss pedagogical fidelity, mathematical fidelity and cognitive fidelity. Use of technology should not compromise your pedagogical principles or mathematical correctness, and the external representations should facilitate the mathematizing processes. Her teaching was first using Geometer's Sketchpad and Scratch. For the next iteration she used lots of apps etc (for instance from learner.org). The problem is that there is a learning curve for each app or software. Now she has gone back to Geogebra exclusively. (And there are lots of things available at Geogebra Gallery.) Her main challenges is the connection between physical, virtual and mental representations - how much is needed of each - and to facilitate mathematics discourse when each student has a computer in front of them.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Fourthly, Ryan Fox discussed "Pre-service elementary teachers generation of multiple representations of word problems involving proportions".<span style="mso-spacerun: yes;"> </span>He used Ball and Rowland for background literature. He stressed the difference between the way PSTs were taught in school and the way they will be expected to teach. (This will certainly be the case - no PST can foresee which school cultures they will work in during their careers.) This study was included four participants, all in their second year of teacher education. They were interviewed four times, totalling 2-3 hours per participants. The interviews concerned how the PSTs would teach certain topics. His students mostly went for one solution method - one struggled a lot, finally found a solution, stuck with that but forgot it by the next interview, another never struggled, but kept using the same solution method. But the fourth student managed to use four representations for the same problem.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><span lang="nb"><br /></span><span lang="nb-NO">After lunch, the TSG continued with two more talks. Gabriel Huszar talked on "An exploratory study about the reseponses of the prospective primary teachers using the concepts of measurement in maths". He started by referring to PISA and other studies, showing the need for intervention in Spanish education. This project tried to identify the most common conceptual problems and mistakes among PSTs in measurement. 81 students took part in the study, and the measurement instrument was a questionnaire of 12 tasks taken from PISA and PIAAC. Sadly, it was hard to see the graphs with the results in this talk. Of course, tasks which does not only ask for a repetition of algorithms, were more difficult for them.</span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then, Mine Işıksal-Bostan talked on "Prospective middle school mathematics teachers' knowledge on cylinder and prism: generating definitions and relationship". This was a video presentation, probably due to the difficult situation in Turkey, due to mr. Erdogan, although this was not mentioned. She discussed the importance of definitions in teacher knowledge. She referred to Tall&Vinner, and Zazkis&Leikin. In this talk, the focus is cylinder and prism. Of course, the formal definitions are different from the definitions used in textbooks in primary school. She gave examples of how cylinders and prisms are introduced in Turkish textbooks. The project were based on questionnaires and interviews. The answers were catagorized according to whether they gave sufficient and minimal conditions (she did not explain why this is an interesting analysis to do - for primary school it may be more interesting that the definition is understood and usable than that it is minimal). They were also analyzed for prototype/non-prototype and other characteristics. In conclusion, the concept image seemed more "correct" than the concept definitions.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(Could I create an activity in which students get a long list of definitions, in which they discuss what are the differences between them?)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After another short break, there were five more presentations in the TSG. My colleague Annette Hessen Bjerke talked on "Measuring self-efficacy in teaching mathematics". She noted that mathematics teaching self-efficacy is particularly interesting, as many express low self-efficacy in maths teaching. She refered to Bandura's concepts of outcome expectancy vs. personal self-efficacy - her research looks at personal self-efficacy in tutoring children. She developed a new instrument for this, with 20 items, 10 concerning "rules" and 10 concerning "reasoning". She mapped self-efficacy through the compulsory mathematics course in Norway. Rasch analysis was used with an instrument that had been validated. She described how it can be shown that the PSTs' self-efficacy improved, but also that there are some students that have decreased self-efficacy at the end of thr course. Further research (ongoing) will investigate the sources of self-efficacy.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lenni Haapasalo discussed "Assessing teacher education through NCTM standards and sustainable activities". He had developed instruments to look at self-confidence of different groups of students, and showed how they developed. However, in the short time, it was difficult to understand details of the instruments, whether they had been validated and so on. He added some comments on the sad state (according to him) of the Finnish educational system and how his colleagues (still according to him) earned money by travelling around the world talking about how good the system was. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Siyin Yang had a talk titled "A comparison of curriculum structure for prospective elementary math teacher programs between the United States and China." She studied two different elementary educational programs. She described the different structures of the programs she had chosen to look at. Of course, the details of the comparisons must be looked up in her papers.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Gabriela Valverde Soto's talk was "Enhancing the mathematics competencies of future elementary teachers: review of a design research". Her chosen topic was ratio and proportionality. She described the phases of design research. This was a rich presentation from a big research project including more than a hundred audio recording, and the analysis looked for productive exchanges, among other things. Again, I can't try to summarize.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Finally, Oleg Ostrovskiy talked on "Visual representations of word problems". External representations can help us solving word problems partly because word problem solving correlates with working memory. He pointed to the Singapore method for solving word problems. The project involved 17 PSTs, using the book "Elements of modelling". Pre-tests and post-tests showed that more PSTs used visual representations to solve the problems, and more adequate ones, after the intervention than before. However, it takes a long time to learn these ways of illustrating - it is not enough to just show the students the method a few times.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">In all, this was 11 TSG-related talks in one day, most of them ten minutes long, therefore quite hurried while still missing out on significant points of the projects. I'm sceptical of the ten-minute format, even though they can of course be seen as a huge number of samples to inspire us to read the papers (even though the papers are only four pages long, most of them, so they are also ridiculously lacking in detail). </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After this, there were another seven hours of partly mathematics-related discussions before I got back to my hotel, but I won't try to summarize these hours later, important though they are.</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-86914205942211974402016-07-25T13:17:00.000-07:002016-07-25T13:17:07.776-07:00ICME13 Day 1 #icme13<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The first day of ICME13 (The thirteenth international congress on mathematical education) was Monday 25th of July, 2016. This conference takes place in Hamburg, Germany. As usual, I will try to blog from the conference (as I noted yesterday).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I missed the conference opening ceremony, but got there in time for the plenary panel on "international comparative studies in mathematics: lessons for improving students' learning". (Not that difficult, as the conference was at that time 40 minutes behind schedule.) The panel participants were Jinfa Cai, Ida Mok, Vijay Reddy and Kaye Stacy. The panel was based on a paper to be published (open-access) shortly.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The panelists each talked on one lesson to be learned about international studies, defined as studies involving at least two "countries", with an intention to compare them. Such studies are good to understand ourselves and to understand different possibilities. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lesson 1: examining the dispositions and experiences of mathematically literate students. Kaye Stacy talked about large international comparisons, PISA in particular. She showed how PISA 2012 were trying to unpack the scores by looking at the processes of doing mathematics. Different countries had different patterns - for instance, Japan had a high score on formulating, lower score on employing and interpreting. Netherlands and the UK had very different profiles. (English-culture countries were best in interpreting.) This fitted not very well with students' answers about their exposure to different kinds of tasks. She also showed graphs showing huge gaps in confidence between boys and girls in some countries (i.e. Australia, but not Shanghai). We need to draw evidence from different sources to inform policy.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lesson 2: Understanding students' thinking. Jinfa Cai talked about small-scale studies, trying to understand students' thinking when trying to solve tasks. He showed two tasks, one where average is demanded, the other where the average is given. About a fourth of students in both countries got the first task right and not the second task.<span style="mso-spacerun: yes;"> </span>He showed some of the different errors in the second task - most concerned adding and dividing, as in the (correct) solution of the first task.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">He also showed the "pizza ratio problem", where he also gave different solutions, some of which were almost only used in one of the two countries studied. Thus, comparative studies gives a broader sense of possible ways of thinking.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lesson 3: Changing classroom instruction. Ida Ah Chee talked about TIMSS Video Study and Learner's perspective study. They have complimentary roles. TIMSS Video Study showed that teaching was a cultural activity. LPS compares lesson structures and lesson events and looks also on practitioners' thoughts about their lessons. Lesson events are as different as lesson structures. Analysing structure and events in light of teacher interviews give an even richer picture than the TIMSS Video Study did.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Lesson 4:<span style="mso-spacerun: yes;"> </span>Making Global Research Locally Meaningful: TIMSS in South Africa. Vijay Reddy talked about the sociopolitical perspectives. She stressed how big differences is a fruitful ground for research. In the Apartheid system, bad education in mathematics for large groups was explicit policy. After 1994, improved access to education has been a priority, but improvement is slow. Mathematics results is a proxy for equality. She discussed how TIMSS can be used to understand the local development. South Africa had big inequalities in the TIMSS scores, reflecting social inequalities. Finally, in 2012, results are better. TIMSS also shows that the level of violence is higher on SA than in any other country.<span style="mso-spacerun: yes;"> </span></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The "panel" did give some ideas for how to use international studies for better understanding, and there are surely lots of data out there that can be looked at, as an alternative to always collect your own data...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After lunch, I attended the talk by Emma Castelnuovo-Award awardees of 2015, Hugh Burkhardt and Malcolm Swan (in absentia). It is particularly interesting to see proponents of what we in Norway call "utviklingsarbeid" awarded and given prominence at such a conference. Their work have resulted in many beautiful tasks and sets of tasks, carefully designed to facilitate particular learning outcomes.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">First he gave information on ISDDE, which has goals of building a design community, raising standards, increase influence on policy. Educational Designer is a journal and the website is isdde.org.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The talk was very inspiring, but difficult to write notes about, as many of the points mentioned were illustrated beautifully with examples of tasks and student responses. But here are a few points:</div><ul style="direction: ltr; margin-bottom: 0in; margin-left: .375in; margin-top: 0in; unicode-bidi: embed;" type="disc"><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">The Shell Centre wanted impact through useful materials, with focus on design. Materials need to fit the system they aim to rearch. It doesn't help if materials are brilliant if they don't work. Therefore, engineering research involves design research and systematic development.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Tasks can provide 'microworlds' for learning.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">They talk about novice tasks, intermediate tasks and expert tasks. Expert tasks involves complexity and unfamiliarity, which means that the technical demand cannot also be high. He gave many examples of the three levels of tasks.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">The usual role of a teacher is to manage, explain, set tasks. Certain tasks facilitates "role shifting", and role shifting changes the classroom culture.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">In general, what this team wants to achieve is technical fluency, conceptual understanding, strategies, appreciation.</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: Calibri; font-size: 11.0pt;">Mostly, lessons that they develop are either concept focused or problem solving focused. Concept focused lessons are developed in the "diagnostic teaching" way. He gave examples of different kinds of the two kinds of lessons.</span></li></ul><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">This talk was a whole lot more inspiring than what these points suggest. He suggested making small units of lessons so that a teacher could substitute their textbook for a few weeks, without having to make a bigger commitment. I would be interested in joining such an effort...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then there was a discussion on a survey on competences, led by Mogens Niss, joined by Regina Bruder, Nuria Planes and Ross Turner. The main issue is "What does it mean to master mathematics?" (Which also begs the question what we mean by mathematics.) We may talk about the products of mathematics or about the enactment of mathematics. Knowing and doing is not the same thing. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Classically, the answers were given in terms of content and related facts and procedural skills. Criticised in Spens report 1938, by Pólya in 1945 and so on. The first IEA study in the 1960s included several components. Also, Papert in the 1970s commented. Since the 1980s, there has been a trend focusing on enactment.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Ross Turner asked what are the relationships between mathematical literacy, numeracy, quantitative literacy, mathematical competence/competencies, mathematics. How would a Venn diagram look like? Or, if you put them in a diagram with products and applications on two axes, how would it look? (As usual, I'm trying to figure out where the history of mathematics and other integral cultural parts of mathematics fit into such a diagram - I think it shows that the concepts here are too narrow. - at least, to "products" should be added "background". Turner, however, put history into applications.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Regina Bruder talked about two types of research: research where the construction competencies are an object and research where it is a means. One important discussion is whether it makes sense to discuss mathematical competencies without specifying which domains they are related to. Also, it is discussed if affective considerations should be included. Also, can they be detected empirically? In general, yes, but they are often overlapping and are not developed or enacted in isolation.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Mogens Niss and the others then discussed how competencies have played roles in different national curricula. Nuria Planes explained that in Spain, competencies have had a big inpact on paper but not in actual implementation and practice. In Latin America, processes have been included in curricula. Ross Turner told us that in many countries in South East Asia, doing mathematics is clearly included in curricula. But of course, the main question is how these words are implemented. Regina Bruder told much the same story for Germany. In Austria, personal and social competencies are included.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then, challenges to implementation were discussed, but this seemed to be the usual list of points. The conclusion was that bridges between action and research is needed - which ties this panel nicely to the previous talk - research and development (in Norwegian: FoU) both are needed, and need to work together.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">That concluded the first day of ICME 13. A day with a varied scientific programme, with a big delay in the morning and with serious temperature problems in the afternoon (the buildings obviously not built for full auditoriums in late July). My<span style="mso-spacerun: yes;"> </span>choice of outfit (shorts + t-shirt) will be repeated for the days to come.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">And as usual, most important is what I do not mention here, lots of talk with colleagues over coffee, lunch, beer and dinner and in-between.</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-2319869227921632092016-07-24T02:47:00.000-07:002016-07-24T02:47:43.968-07:00ICME13 Day 0The welcome reception for <a href="http://www.icme13.org/" target="_blank">ICME13</a> - the 13th international congress of mathematics education - is today. I'm flying to Hamburg tomorrow (Monday morning) and misses not a lot. This will be my fifth ICME, having participated in ICME9 (Tokyo, Japan), ICME10 (Copenhagen, Denmark), ICME11 (Monterrey, Mexico) and ICME12 (Seoul, South Korea). I hope I'll also go to ICME14 in Shanghai four years from now.<br /><br />As usual, I'll try to blog from the conference. As usual, the blog will contain my understanding of what I hear, as well as some thoughts I get from what I hear - it is not trying to give a completely fair and accurate account of the presentations. For that, please consult the actual papers...<br /><br />Here are some of my blog posts from previous ICMEs:<br />2012: <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-0.html" target="_blank">Day 0</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-1.html" target="_blank">Day 1</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-2.html" target="_blank">Day 2</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-3.html" target="_blank">Day 3</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-5.html" target="_blank">Day 5</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-6.html" target="_blank">Day 6</a> - <a href="http://teachereducatorbjorn.blogspot.no/2012/07/icme-day-7.html" target="_blank">Day 7</a><br />2008: <a href="http://bjornsmaths.blogspot.no/2008/07/icme-day-1.html" target="_blank">Day 1</a> - <a href="http://bjornsmaths.blogspot.no/2008/07/icme-day-2-part-1.html" target="_blank">Day 2 (part 1)</a> - <a href="http://bjornsmaths.blogspot.no/2008/07/icme-day-2-part-2.html" target="_blank">Day 2 (part 2)</a> - <a href="http://bjornsmaths.blogspot.no/2008/07/icme-day-3.html" target="_blank">Day 3</a> - <a href="http://bjornsmaths.blogspot.no/2008/08/icme-day-4.html" target="_blank">Day 4</a> - <a href="http://bjornsmaths.blogspot.no/2008/08/icme-day-5-part-1.html" target="_blank">Day 5 (part 1)</a> - <a href="http://bjornsmaths.blogspot.no/2008/08/icme-day-5-part-2.html" target="_blank">Day 5 (part 2)</a>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-5772926174177627482016-06-02T00:57:00.001-07:002016-06-02T01:00:52.318-07:00Conference on diversity in academia (part 1) <div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I am happy that HiOA organizes a conference on "mangfoldsledelse i akademia". This is a very important topic to succeed as a university, and in particular it is important not to have a too narrow idea of "diversity" and also not to reduce diversity to counting. Of course, the topic is also important to the HiOA board. (Sadly, I missed the first few minutes of the conference because of other obligations.) As usual, the notes below are my on-the-spot interpretation of the speakers' points, and does not necessarily reflect what they tried to convey.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(Just for fun, I had a look at the list of participants. Of 128 names, 24 are male-looking. That's less than one fifth. It's harder to count the number of white or the number of straight people by looking at the list, by the way...)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The first talk, after the rector's introduction, was <b>Ronald Craig</b> from <a href="http://www.ldo.no/" target="_blank">LDO</a>. He talked about the importance of "critical mass" - how prejudice evaporates as a previously underrepresented group reaches "critical mass". </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Most important components to get progress:</div><ul style="direction: ltr; margin-bottom: 0in; margin-left: .375in; margin-top: 0in; unicode-bidi: embed;" type="disc"><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: "calibri"; font-size: 11.0pt;">What are the specific barriers to equality (look for risks, not actual instances of discrimination)</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: "calibri"; font-size: 11.0pt;">A demonstrated top management commitment</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: "calibri"; font-size: 11.0pt;">Management accountability</span></li><li style="margin-bottom: 0; margin-top: 0; vertical-align: middle;"><span style="font-family: "calibri"; font-size: 11.0pt;">The setting of numerical goals and timetables</span></li></ul><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(In a break, I was asked by someone if there was any other area where these same four points do not apply. I'm not sure.) Diversity training does not show effect in research, having mentor programmes or a diversity taskforce does show an effect. (Kalev & Dobbin)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Next up was <b>Mary Ann Danowitz</b>, professor and dean at NC State University in the USA. She started by saying that academic institutions are different than companies, and we have to be careful about drawing on experiences from companies. She reiterated the well-known reasons for focusing on diversity management (in the sense of non-discrimination), most importantly (perhaps) that the whole idea of the academic life is that ideas should be judged on their merits, not on the basis of a persons' gender etc.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She mentioned important mechanisms that creates differences, such that women marrying and having children is correlated with not being tenured, and that women tend to become more isolated and have to work harder to be respected in the field.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She enumerated four dimensions of diversity management (DM): valuing diversity, strategically strives, initiatives and structures, focus on added value. There is a lack of comprehensive work where all these dimensions are included.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">To hire and retain a diverse academic staff, several ideas are good: communicate a commitment to diversity and equality, broaden the professional networks of faculties, reconsider the role of diversity in lists of qualifications, consider dual career opportunities (that is; jobs for both partners in a marriage/partnership), consider demands on and possibilities for developing Norwegian language skills.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Curt Rice asked how to change the whole institution, including the academic staff. Danowitz answered that what the academic staff is primarily interested in, is research and their courses. They need to see added value for their research and courses.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After a break, <b>Julien Bourelle</b> from NTNU talked on "An international perspective on Norwegian academia". He had an entertaining talk on cultural differences. For instance: the Norwegian concept of politeness stresses not disturbing others, while in many other cultures, this (not saying "Hi" to everybody all the time) can be interpreted as being impolite. Social bubbles are important - in Norway, it is difficult to get invited to anything unless you take part in activities (sports, chess, ...) People from other cultures are more used to direct feedback/praise. Norwegians also enjoy huge personal space. If people move away from you at a bus stop, it is easy to believe that it is because of skin color, while it may in reality be just because there is more space somewhere else.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">It is important to see the world not only through our own lense, but also through the lense of other people's cultures. We need to see how other people see the world, and that is easier when we are diverse ourselves.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Julien is behind the Facebook site "<a href="https://www.facebook.com/LoveNorge/" target="_blank">I fucking love Norway</a>" and wrote "<a href="http://scandinavian.tips/products/the-social-guidebook-to-norway" target="_blank">The SocialGuidebook to Norway</a>".</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">That concluded the pre-lunch part of the day. The rest of the day was supposed to be in Norwegian, and then it's easier for me to comment on that in Norwegian (<a href="http://larerutdanneren.blogspot.no/2016/06/konferanse-om-mangfoldsledelse-del-2.html" target="_blank">in my Norwegian-language blog</a>). </div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-26234499439833356902015-11-26T22:46:00.000-08:002015-11-27T00:20:36.129-08:00Hurray! The quality of my article just increasedOne way of measuring the quality of research and development projects are through "impact factor" (that is, how many references do an article get). Therefore, I'm always happy when I get an email alerting me that someone has referenced one of my articles - because that means that the quality of my article has increased one notch.<br /><br />A few days ago, for instance, I got an email from Google Scholar telling me that my eminent article "Teachers' conceptions of history of mathematics" had been referenced by the scholar Suphi Önder Bütüner in the article "Impact of Using History of Mathematics on Students’ Mathematics Success: A Meta-Analysis Study". It is nice to be referenced, especially in the rare event that it's not me or a close colleague who is the author of the referencing article. (Although maybe the author and I had a colleague in common.)<br /><br />So which of my words of wisdom was picked up by Bütüner, thereby proving the quality of my research? The article is, as the title suggests, a meta-analysis, and - to make a short story short - it turns out that Bütüner has read my article and concluded that my article is about history of mathematics but is not focusing on the pupils and their learning of mathematics. Therefore, my article is mentioned in a list of articles that are relevant to the field of study as a whole, but not to the specific problem that Bütüner wants to investigate. Therefore, my article is mentioned as one of the article not useful for the analysis.<br /><br />Nonetheless, the quality of my article is higher than one week ago. In other cases, I have seen references to my work, but the researchers have cited me as saying something that I was quite unable to understand how the researchers could possibly think I was saying. Thus, the quality of my article (as measured by impact factor) have increased, even though it is patently impossible to understand what point I was trying to make.<br /><br />Let me end by pointing out that the fact that impact factor is a quite imperfect way of measuring scientific quality, does not suggest that the system of counting articles used in Norway (with different points given based on the journals published in) is better. (As it happens, different articles in the same journal have an annoying tendency not to be of exactly the same scientific quality.)<br /><br />(This blog post is a translation of <a href="http://larerutdanneren.blogspot.no/2015/11/hurra-kvaliteten-pa-forskningen-min-kte.html" target="_blank">a post in my Norwegian blog</a>.)Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-41154589032301669902015-03-05T21:51:00.003-08:002015-03-05T21:51:34.152-08:00NERA 2015 Day 2 #NERA2015GUT<span style="font-family: Calibri; font-size: 11pt;">he second day of NERA started off with parallel sessions, I chose the one on "Teacher's work and teacher education". First, my colleagues Kirsten Thorsen and Hanne Christensen talked on "Identity forming in teacher education". They are part of a project TPQ (as am I), and their talk is based on data from two of the sub-projects. The data involve surveys</span><span style="font-family: Calibri; font-size: 11pt;"> </span><span style="font-family: Calibri; font-size: 11pt;">and in-depth interviews with students, mentors and campus teachers. Students describe practice as the terrain - authentic and unpredictable - in particular in the first year. They describe campus learning as the map. The project started out with looking for "the gap" to be able to bridge it - but the "gap" metaphor doesn't really work if it is the terrain and the map. Peers are very important in both areas. The practice mentors are seen as giving solutions, while teacher educators give them theory.</span><br /> <!--StartFragment--> <div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Practice teachers do not feel competent in theoretical themes, so they do not connect practice to theory very much. They tend to focus on practical advice and discussions on what works in practice.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Secondly, Roald Tobiassen had a talk on "Portfolio as practice in teaching practicum: promoting reflection and constructing teacher identity". The project was connected to the teacher education that is called PPU in Norway, and a part-time model. The portfolio was meant to scaffold students' practicum learning and reflection. During the first year, the students had six pedagogical texts to write, starting with "My pedagogical creed". The students meet in groups of 4-6 students which discuss the tasks. This particular project looked at six of the students.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Tobiassen went quickly through a lot of the theory on portfolio. Then to the findings: these six students were positive with regards to the portifolio - they saw it as a way of developing teacher identity - they valued the authenticity. "Portfolios helped me to see where I wanted to go and how to get there". The six students were happy with the structure as well. They saw the portfolio as a tool for connecting what they read about being a teacher and their own experience.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Finally, Jóhanna Karlsdóttir's theme was "Storyline som metode i inklusiv læring og undervisning i praktik hos lærerstuderende". (Even though the title is in Swedish, most of the words are understandable for the English reader, I guess...) The study is based on one course in teacher education in Iceland, with 21 participants (in 6th semester) - the data are interviews, notebooks, discussions etc. The course is focused on inclusive education, which is apparently not taught systematically in teacher education in Iceland. She presented an eclectic mix of theories as a foundation for her work, including Gardner, Johnson & Johnson etc. She then described the storyline process (in the classical way). Her project goes on for one more year, but she has found some room for improvement, for instance in getting to know the method better before using it themselves. But storyline apparently is useful for inclusive education as it is building on pupils' resources.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I do think that I see more of why I do not like such a "broad" conference as much as more focused conferences. I think the variety of participants makes it almost impossible for those giving talks to present their projects effectively - it is impossible to know which theories and concepts and contexts are well-known to the people present. Thus, most presentations either spend too much time on stuff I already know well or jump too quickly over theories I do not know. This is in marked contrast to for instance the HPM conferences I go to every four years.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Jane Kenway, Monash University, had a plenary on "The emotional life of markets in education". She presented a project looking at elite schools around the world (but not generalizing educational policy based on these schools like yesterday's speaker). Elite schools may be fee-based or merit-based (and grant-funded). She talked on concepts such as "emotional geography" (how do feelings connect to places) and "economies of emotion" (the emotions of buying a Nike shoe is not connected to the smell and sound of the shoe factory). What emotions are evoked by certain schools to make parents send their pupils there? </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">She mocked the way elite schools market themselves with cliche slogans. (It does remind me of a student saying that she was so amazed of how teachers in her school in Uganda always reminded their pupils that they are the future of Uganda. Cliche, yes, but still an important reminder for the children. Isn't there too little - rather than too much - talk of the importance of schooling in western societies?) </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">While the subject of elite schools is a bit interesting, to me it feels like it is on the edge of what I'm interested in. I'm far more interested in how similar mechanisms are working (or not) when "ordinary" schools are concerned. How do competition for pupils influence the internal life of "ordinary", "non-elite" schools. Perhaps ideas for studying this can be found based on the study of the elite schools? Other characteristica of elite schools are certainly less interesting in that context, for instance how parents use relocation services to set up meetings with potential schools - not a very common practice among parents relocating within Oslo, for instance. The high pressure for performance in elite schools - the fail anxiety - is also something that is far less usual in "ordinary" schools, I would think.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">A somewhat relevant feature of elite schools is the development of new "departments" tasked with producing emotions, for instance marketing departments. This is also the case for many Norwegian institutions, for instance my own, with its "avdeling for samfunnskontakt" which is trying to give HiOA a good image and avoid bad press. (While trying to remember that an important part of being a good institution in higher education is to encourage discussions and enjoy the benefits of free speech, even when uncomfortable.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After lunch, I went to the parallel session on Classroom Research. Ingvill Krogstad Svanes presented "Teachers' instructional practices during students' individual seatwork in primary school". Her research is on six teachers in 3rd grade in Norwegian - one week each. She presented an analytical framework developed through the project. The main codes were instructional support, organizational support, emotional support, monitoring and "no direct interaction with students". One main finding is that there is a huge variation between the teachers in how they spend their time. Two of the six teachers give more instructional support than anything else. Others spend most of their time on organizational support. This seem connected to the clarity of the initial instructions and the choice of activities (scissors and glue lead to more need of practical help, for instance). Emotional support was almost not present, but that may be because only what was spoken was coded. Further subdivisions in the codes show that even within the categories there are important differences - some teachers are mostly telling the pupils, while others are challenging them more. (For the sake of openness: I am Svanes' boss as well.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Next, Malin Norberg talked about "How do children in primary school make use of illustrations in mathematics textbooks?" She chose subtraction as an example (an interesting choice, as it is fairly difficult to illustrate in a static picture). 1742 illustrations from 21 textbooks were the data, in addition to discussions with twelve students about five illustrations. She has looked at two subtraction situations: decrease and compare. 86.5 % of illustrations were illustrating decrease. Sometimes the students need the illustration to do the subtractions, in other instances the illustrations are just illustrating a process.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Students sometimes read more into the illustrations than intended, and sometimes they are able to solve the mathematical task with symbols but not with the illustrations. (Not all illustrations are very good...) The teacher's role is important. In her PhD, she will work on teachers' role and teacher's guides.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">For the last scheduled talk in this session I decided to just listen instead of writing notes - sorry...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After a coffee break, I went for the parallel session on Gender and Education again. The first talk was supposed to be by Jenny Bengtsson and Eva Bolander: "What the school can (not) do: Education markets and negotiation on sex, risk and schooling". However, this talk was cancelled...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The second talk of this session was Chie Nakazawa's "Adolescents' norms, attitudes and values regarding sexual and reproductive behaviours from a gender perspective - a comparison between Japan and Sweden." The talk is based on surveys of university students, about 500 in Sweden and about 2500 in Japan. One indication of the difference: about a quarter of Japanese women in the study claim never to have been interested in sex, while only 3.6 per cent of Sweden women claim the same. I am quite sure that this is a cultural and not a biological difference...</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Also, only 2.4 per cent of Swedish men say they are homosexual or bisexual, compared to 10.4 per cent of Swedish women. This question was not even asked in Japan. Moreover, it is clear that Swedes find sex to be more enjoyable and less dirty and embarrassing than Japanese, if the survey answers are to be believed. Also on gender roles, the differences are significant.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thirdly, there was the talk "A hotbed of heterosexuality? On the reproduction of notions of sexuality in language instruction" by Angelica Simonsson. Her talk is based on her ongoing PhD project. Her research question is whether sexuality and gender normativity is constructed in language education instruction in secondary school. She is looking both at teaching materials and at how pupils and teachers relate to these. It is based on two classes (grade 8) in two different schools, and the teaching aids used. Subjects: Swedish and English.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Her findings are that sexuality figure in teaching aids, and these are all hetero. This goes for both non-fiction films, fictional films, short stories and textbooks. There was a total lack of non-heterosexual relationships. (This is quite surprising, as my research show quite a bit of homosexuality in Norwegian textbooks. Does this mean that Norwegian textbooks are more inclusive, or would a study of Norwegian classrooms show a similar pattern, i.e. that teachers choose the heterosexual examples?) (in a comment at the end of the session, it was argued that even when gay characters are portrayed in literature for youth, they are portrayed in a very stereotypical way. I have not looked at that in my material.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Finally, the fourth talk of the day was Per Nordén's "First Generation Rainbow Children Speak Their Minds - How queer kinship structures matters in education". With "rainbow children", he refers to children with one or more LGBT parent. The talk is based on interviews with 28 rainbow people from age 15-37. He gave long quotes from different stages of history. It is fascinating to see how different experiences are, and that these are intimately connected to changes in law and society. Sadly, he did not have the time to come to the part of their school experiences. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">That's the end of the second day of NERA. Time for the last preparations for Friday morning's talk... </div><!--EndFragment-->Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-87416366695833919842015-03-04T10:58:00.001-08:002015-03-04T10:58:29.803-08:00NERA 2015 Day 1 #NERA2015GU<div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">I will "report" on my first NERA conference in the same way as I report on other conferences I attend - through quick notes written throughout the conference. The conference lasts for three days, and I am having a presentation on Friday morning (and <a href="https://m.youtube.com/watch?v=Wz67irbypkI" target="_blank">an early version is available on YouTube</a>...) Hesitatingly, I will do the notes in English, as most of the presentations will be in English, even though most participants are Swedish and Norwegian.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">After registration and lunch, the first thing on the programme was the opening, of course. Dean Roger Säljö welcomed us and gave a brief introduction to the Faculty of Education at the University of Gothenburg. There was also an introduction into the research activities in the faculty.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The first plenary talk was by professor Hugh Lauder, titled "The Repositioning of Education in the 21st Century and what Can Be Done About It". His starting point was the tightening bond between the economy and education in the 20th century - which he claimed is now eroding. In some countries, education's only goal (in political documents) is the economy. It has been assumed that education leads to upwards mobility, but that is dependent on an increasing number of jobs at the top (or downwards mobility on the top). Education is also seen as important for global citizenship, but this is dependent on children seeing a future, which they don't always see.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Now, there is an increasing polarization in wealth and income, and there is an increasing competition for a decreasing number of top jobs - the corporate ladders are now flat. We see the emergence of a global education system which educates multi-lingual multi-cultural candidates for transnational corporations. Those not "talented" are hitting a glass ceiling. (The argument is unconvincing, as education could certainly be economically worthwhile even though it does not give top jobs in multi-national corporations...)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">32 percent of the poorest 10 per cent of British people are graduates - meaning that graduation is not guarantee against poverty. An increasing percentage of graduates are poor. (But again, this does not mean that education is not worthwhile for the individual. And it certainly doesn't mean that education is not good for the economy as a whole.) There is an insufficient supply of high skilled work.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">(By the way, Lauder is the kind of lecturer who has a Powerpoint with huge amounts of text which he shows as he is saying something else. This doesn't work very well for me, maybe because I'm not a native English speaker and reader...)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">He ended by saying that we cannot claim that education is there for the economy, what is then the purpose of education and how can we then get funding for it? (But of course humankind have discussed for thousands of years what education is for, so we do have plenty of non-economic answers to that question.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">As I have said, I don't find the arguments in this talk convincing - the numbers don't seem to add up to the conclusions he is stating. The data are on the outcome for the individuals, not for society. Perhaps the reasoning is better explained in his book(s). (But as was commented later in the day - in that case it would be good if he put his best arguments into his talk.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then, after a coffee break with a conspicious lack of coffee, there was the first parallel sessions. I went to the ICT & Education session, where three of my colleagues from Oslo presented. But first, Ann-Katrin Perselli had a talk entitled "From computer room to one-to-one". She described a phenomenological study with four teachers from two upper secondary schools, in which all students had their own computers. Among her findings: each student having a PC meant less fighting for the computer lab, while PCs were also disturbing. Teachers based their teachings on tips from colleagues, trial and error and websites - this was time-consuming for inexperienced teachers. Good relationships with students and teachers were a help. The study apparently shows how teachers' "lived experience" influence their approach to using IT in teaching. School leaders need to be aware that teachers are different.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Then, Bård Ketil Engen and Louise Mifsud presented work on an online course on collaborative learning activities - on master level. The course has been held for four years. They discussed different ways of engaging students online, now using Adobe Connect and Second Life. Asynchronous student collaboration was mediated via Etherpad and Wiki. The semester is designed with student activities alternating with online synchronous lectures, and finally there is an exam where students write a paper on the work they have been doing.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Technology influences communication. Students often get more passive online, and maybe a bit uncertain. Asynchronously, we see that students start out cooperating (dividing the labour) instead of collaborating. Overall, students are learning about CSCL activities while learning about CSCL. (I happen to be the boss of these fine teacher educators, a fact there's no reason to hide.)</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Finally, Marianne Vinje had a talk with the title "Teacher Strategies for Meaningful Learning in a Blended Environment". Many challenges are facing higher education - resources are moving away from teaching, and research is rated higher than teaching. The role of instruction gets less important, teaching complex (higher-order) thinking is more important. Technologies give endless opportunities which have to be developed. One such opportunity is blended learning, which is what Vinje has been working on, using a community of inquiry (CoI)framework. </div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Teaching online is something else than traditional teaching - other factors are important than in traditional teaching. Studies show a change of many teachers from an instructional mode of teaching to a more Socratic mode of learning. Also, many teachers are more precise in their messages/information. Vinje thought blended learning gave her more classroom time to get to know her students.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">As the last stop of the day, I chose the parallel session on Gender and Education. The first talk here was by Ingólfur Ásgeir Jóhannesson on "Does the National Curriculum Guide 2011 pave way for gender and queer studies in Icelandic schools?" The Icelandic National Curriculum has six fundamental pillars, one of them is equality. This talked is based on text studies and interviews of teachers. There is a focus on equal opportunities in the rest of the curriculum, but rarely on gender studies. Sexual orientation is mentioned, queer and queer studies are not mentioned. One of the books analysed was "I, You and We All" (for 6th to 8th grade) in this, intersectionality is clearly used, and there is a social understanding of gender.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Gender studies is an elective course in some upper secondary schools in Iceland. This is a course without a textbook or final exam. Queer studies is not a specific course anywhere, but is a part of gender studies. The inclusion of gender studies as an elective course is the result of a spontaneous movement among upper secondary teachers, supported by student interest. In one school, Pink Holocaust is taught.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">The second part of that parallel session was Anja Kraus' talk on "'Gender' as a Tacit Dimension of Pedagogy". Her starting point was that the the aim of pedagogy is to set people free, the idea of Bildung. Gender can be seen as an analytical tool, helping to understand the constitution of practices and knowledge domains. Traditional pedagogy tends to rely on logic and on concepts that are supposed to exactly fit reality, she argued. Postmodern approaches rely more on self-interpretations. "Bring the body into the discussion" was one expression used. Queer theory, Butler's performativity and body-phenomenological concepts were contrasted, and apparently the latter was the preferred one.</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">As far as I understand it, a text is seen as problematic because it imposes logic on the world, which makes it a bit problematic to give a talk (a text) on a postmodern approach (or on anything, really).</div><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;"><br /></div><!--StartFragment--> <!--EndFragment--><br /><div style="font-family: Calibri; font-size: 11.0pt; margin: 0in;">Thus ended the first day of NERA 2015. A very varied day - a bit more varied than I prefer, I guess. But I do bring some interesting points with me from this day.</div>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-80372779960553440962015-01-01T01:24:00.000-08:002015-01-01T01:24:41.495-08:00I'm not happy with the way I'm referred to...Some time ago, Google Scholar noted that there was a reference to me in a new paper, <a href="http://seahipub.org/wp-content/uploads/2014/07/IJIER-J-5-2014.pdf">The Need for the Inclusion of History of Mathematics into Secondary School Curriculum: Perceptions of Mathematics Teachers</a> by Habila Elisha Zuya. It is published in the journal <a href="http://www.ijier.net/faq.html">International Journal for Innovation Education and Research (IJIER)</a>, a journal that is peer-reviewed, but states that "The review process takes maximum two weeks." Hm...<p> The article refers to me in this way: <blockquote>A number of researchers have pointed out that teachers' interest in mathematics increased when introduced to the history of mathematics (e.g. Smestad, 2009; Siu, 2004; Phillippou & Chritou, 1998; Stander, 1989). However, these researchers maintained that teachers found no interest in using the history of mathematics within the curriculum. </blockquote> I do not think my data can be used to claim that "teachers' interest in mathematics increased when introduced to the history of mathematics" or that I claim that in the article. Neither can my article be used to suggest that "teachers found no interest in using the history of mathematics within the curriculum". Of course, I may be wrong, maybe indeed my article does suggest something else than what I intended. But I am more tempted to believe that this is a case of trying to fit too many references into too short a paper, so that the actual point of view of each reference is not retained.<p> I just wanted to get it off my heart. I don't think the journal would be interested in publishing a note to this effect, at least not without me paying for the privilege...Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-66238537918072581992014-10-27T04:55:00.003-07:002014-10-27T04:55:37.222-07:00Comments on Gert BiestaToday, professor Gert Biesta was visiting HiOA, He presented his article "How does a competent teacher become a good teacher? On judgement, wisdom, and virtuosity in teaching and teacher education." (in R. Heilbronn & L. Foreman-Peck (Eds.), Philosophical perspectives on the future of teacher education. Oxford: Wiley Blackwell.)<br /><br />As part of the seminar, I was asked to prepare a comment from the point of view of Norwegian teacher education. The seminar was not filmed, but I have made a video with my comments:<br /><div class="separator" style="clear: both; text-align: center;"><object width="320" height="266" class="BLOGGER-youtube-video" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0" data-thumbnail-src="https://ytimg.googleusercontent.com/vi/uVPC_tAlKco/0.jpg"><param name="movie" value="https://youtube.googleapis.com/v/uVPC_tAlKco&source=uds" /><param name="bgcolor" value="#FFFFFF" /><param name="allowFullScreen" value="true" /><embed width="320" height="266" src="https://youtube.googleapis.com/v/uVPC_tAlKco&source=uds" type="application/x-shockwave-flash" allowfullscreen="true"></embed></object></div>My "manuscript" can be found at <a href="https://www.academia.edu/8961127/Comments_on_Biesta_2014_" target="_blank">Academia.edu</a>.<br /><br /><br /><br />Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-39805360741053638172014-07-18T12:47:00.002-07:002014-07-18T12:47:53.423-07:00Post-conference blues Last week, I sat in solitary confinement in my office. <br>I struggled with the contents of my workshop - who would care? <br>Then here - the adrenaline made my senses aware<br>of every single sigh, shaking head or smile.<br><br>That done, every waking hour was spent with soul mates -<br>breakfast, lunch and dinner and all the hours in between.<br>People all around the world care about what I do!<br>People share their thoughts and listen to my thoughts, too.<br><br>Then it ends. The plenary room is suddenly silent.<br>The echo of many "see you in two years" fade.<br>Life gets back to normal: and suddenly I see:<br> again, the only one here who care what I do - is me.<br>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-33946625289219845592014-07-18T10:19:00.001-07:002014-07-18T12:10:54.771-07:00ESU7 Day 5The final day of ESU7 started with Bjarne Toft's "Peter and Sylvester - the emergence of graph theory". The first paper on graph theory was by Danish mathematician Julius Petersen. Of course, in hindsight, precursors of graph theory can be seen in The Bridges of Konigsberg problem, for instance. Road networks, social (or <a href="http://researchnews.osu.edu/archive/chains.htm">sexual</a>) relationships and website connections are examples of applications today.<p> Petersen was a teacher, teaching 35-36 hours a week, and publishing a new textbook every other year. He wrote a much used and acclaimed collection of geometrical problems. He finished his PhD in 1871, and published a lot in the following years. Then he got in touch with Sylvester, and they met in Copenhagen (in Tivoli!). In a letter of October 18th, 1889, Sylvester explained to Petersen what "graphs" are. In 1890, Petersen went to England to collaborate with him. The collaboration didn't work out, though, and there were some amusing letters going in several directions. They showed both how mathematicians are human beings with mood swings like everybody else, and that mathematicians are fallible, unlike mathematics books... </p><p> Fàtima Romero Vallhonesta and others then had a "workshop" on "Teacher Training in History of Mathematics" (but it turned out to be more of a lecture). They are a group of teacher making materials for the classroom which will be a publication in two year's time. They had a brief introduction, mentioning the difference between explicit and implicit use of history of mathematics. They pointed out that with teacher students, they used sources explicitly, while with other students, they used the sources mostly implicitly. </p><p> The aims of the implementation was </p><ul><li>Knowledge of the original sources </li><li>Recognition of the most significant changes in the discipline of mathematics </li><li>To emphasize the socio-cultural relations of mathematics with politics, religion, philosophy and culture. </li><li>(The most important). To encourage students to reflect on the development of mathematical thought and the transformations of natural philosophy. </li></ul> There was an introduction to Chinese sources, including Nine Chapters, and then we looked at chapter 9, being given the problem, answer, algorithm and comment (by Lui Hui). (In both the original Chinese and in a French translation...) Students could do activities, for instance doing geometrical proofs on paper. (It was unclear to me, however, what the students were asked to do with the sources and what instructions they were given.)<p> Al-Khwarizmi was then the next example - the usual solving of quadratic equations, but with both of al-Khwarizmi's ways of solving (and this time in an English translation). The goal of this particular task was that students should be better at algebra, and the original source is (as I mentioned) only read by teacher students, not with the mathematics students. First, students are asked to research al-Khwarizmi's life (some basic questions), then they learn both algebraic and geometric methods. The geometric method they learn by getting the complete geometric proof, but without numbers added to the figure, so that they themselves just provide the details, not the steps.</p><p> Their way of using al-Khwarizmi with mathematics students is quite unhistorical (as was pointed out by people attending) and I'm not sure that it could be called implicit use of history; rather it is teaching loosely inspired by history. Teacher students, on the other hand, are given the original sources and given the task of making activities based on them, and then the different ways are compared. I would be careful about doing that with my students, as there is a danger that all of them would make unhistorical materials so that it could be difficult or at least time-consuming to avoid ending the project without having reached anywhere meaningful.</p><p> Finally, there was an example from Pedro Núnez and from Viète. There was a fascinating dispute in the end of the workshop where an equation in the style of Viete was written also in the style of Descartes, where the variable A was changed into the variable x, but the z from Viete was kept. It was pointed out that this would be confusing, as z was a constant with Viete, but a variable with Descartes, so in the "Descartes version" of the equation, it should be c...</p><p> Kristin Bjarnadottir's workshop was a follow-up on her plenary the day before, with the opportunity to go into details. First, we had a look at another problem on measurements, which - among other things - showed Icelanders familiarity with different measurement units, as well as payments based on gender. Then we all collaborated on doing a Geogebra simulation of the movement of the sun in the Icelandic sky at different times of the year. This ended up with the function f(x)= -(90-65)* cos (x*2*pi/360) + a, where a is a value between -23.44 and 23.44 (on a slider), and where 65 is the latitude of the farm of Torsteinn. (Reykjavik 64, Torsteinns place 65, Rome 42 and so on.)</p><p></p><div class="separator" style="clear: both;"><a href="https://lh5.googleusercontent.com/-O4-7QCQLqCo/U8lxOvY5HnI/AAAAAAAAFT4/u_z-5WErif4/s640/blogger-image--1337689808.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://lh5.googleusercontent.com/-O4-7QCQLqCo/U8lxOvY5HnI/AAAAAAAAFT4/u_z-5WErif4/s640/blogger-image--1337689808.jpg"></a></div><br><p></p><p> Then there was a little on the dominical letters, summer's extra week, the first week of summer and so on. I cannot possibly summarize this.</p><p> It was unavoidable that this conference too would end. The closing session was a time for praise for the organisers, but also to look forward, to the deadline for the proceedings papers (10-15 pages by November 15th), to the next HPM conference (Montpellier July 18th-22nd, 2016) as well as to the next ESU (Rethymnon, Greece, July 2018).</p><p> What were the highlights in the week for me? To be honest, the highlights for me personally was a conversation with Renaud Chorlay over a beer after the excursion, a late-night discussion with Francesco Maria Atzeni, breakfast talks with Costas Tzanakis, a few short lunch chats with Torkel Heiede, the traditional dinner with Kristin Bjarnadottir and Andreas Christiansen (traditional since the last ESU...) and coffee break talks with Mustafa Alpaslan. I could go on and on. The most important part of any such conference is the face-to-face discussions with other people who spend their life striving for some of the same goals as yourself, having some of the same interests and strangenesses... Sadly, my blog posts don't catch all of this very well, although my opinions throughout are of course enhanced by the conversations I've had throughout the conference and at previous conferences.</p><p> It is difficult for me to pinpoint a highlight from the scientific programme. Single ideas, such as David's use of the concept of "violence" to describe working with original sources, were thought-provoking. However, if I am to point out two main ideas that have (re)formed in my head during this conference, it is these: </p><ul><li>The same original source can, depending on the context, be used for many different purposes, and depending on these purposes, the design of the teaching will vary. The assessment of whether the teaching was successful, will also vary. Understanding better how these things work constitute major "open questions". </li><li>In particular, in some contexts it is meaningful to give students an original source and a set of tasks to "guide" their reading. When this is appropriate, how these can best be designed and what effect this has on the outcome, form a subset of these open questions. </li></ul> Of course, these questions was actualized already in the first plenary, so I'll send special thanks to Adriano Dematte for his plenary. But I've come across things reminding me of this every day of the conference. <p> In addition, my plan of writing a book for teachers (in Norwegian) on different ways of including history of mathematics in teaching, remains on the to do-list. More resources for teachers are always needed.</p><p> So, as I leave for a few days of holiday in Paris, I know there will be work waiting for me back at work for years to come (and in particular when I'm back in my research and teaching position from 2016...)</p>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-34434800560267669992014-07-17T09:27:00.002-07:002014-07-17T09:27:25.558-07:00ESU7 Day 4Thursday started with familiar faces. First, there was Kristin Bjarnadottir (from Iceland), whom I have met in many conferences before, and whom I have blogged about before, I believe. Even though her field of research is far from mine, I remember being very inspired by her workshop at a previous conference. Now she had a plenary on "Calendars and currency - Embedded in culture, nature, society and language".<p> She started out by talking about ethnomathematics and presenting the history of Iceland. Due to a lack of metal, exchange of goods were the norm instead of buying with money. A hundred was equivalent to a cow or 240 fishes. In 1880: Danish krone adopted. Kristin gave an example of quite involved textbook problems using these exchange "rates".<p> The Misseri calendar was adopted in 930, and this was adjusted to the Julian calendar in the 1200s. The calendar was needed to establish the meeting times of the parliament. The practical problems with errors in the calendar (which meant that soon the parliament had to meet in the busiest parts of the farming year) was a powerful motivation for figuring out how to improve on the calendar. Later, other errors were found in similar ways. But in adjusted versions, the Misseri calendar has survived almost to our day.<p> Secondly, there was Gert Schubring's plenary "New approaches and results in the history of teaching and learning mathematics". He stressed that he avoided the word "education" In the title to avoid confusion with "mathematics education" (matematikdidaktik). He started by outlining the history of research on history of mathematics in school. New trends now: more specialized studies of specific countries, attempts at an international history and new methodological approaches, going beyond the administrative documents and using insights for instance from sociology. These efforts were mostly individual, but have recently also become institutional, thanks to the topic study groups of ICME from 2004 and launching the journal IJHME in 2006. In 2014 was published Handbook on the History of Mathematics Education. <p> If history of school mathematics is reduced to the history of curricula, the social reality is missed. Also, mathematics never occures in an isolated way, but must be seen in connection with other disciplines. Mathematics education became needed by state administrations in the same way that numeral systems and calculation with these were neccessiated by state administrations. Shubring went into some details on the origin of mathematics education in for instance China, which I cannot possibly summarize here.<p> Another question is how and why mathematics turned from something you should learn for state administration and into something everybody should learn. It turns out that France was the first country where this happened, after the revolution of 1789. Latin and mathematics were the two essential disciplines. Again, the amount of detail from different countries cannot be done justice here.<p> The second panel of the conferences had the title "The question of evaluation and assessment of experiences with introducing history of mathematics in the classroom". The panelists were Leo Rogers, Janet Heine Barnett, Ysette Weiss-Pidstrygach, David Guillemette and Frederic Metin.<p> Leo Rogers spent some time outlining the vast canvas we are moving about on: different target groups, different goals, different ways of including history of mathematics... These different contexts we are in, give different answers to questions such as why to assess, what to assess, how to assess, what data to collect and how to analyse and how to respond to the analyses. He also walked us through a lot of the common problems with testing/assessment, which are often of different forms when it comes to history of mathematics than in mathematics. (Of course, research on assessment in mathematics and history could have much input to us here.)<p> Janet talked on her work on the projects she has been involved it (with resources available online). The primary source projects have student tasks, which may be relevant for assessment. The primary goal Here is for students to learn core material in contemporary mathematics courses. But there are also hopes that these original sources support a development of a deeper level of understanding. The examination is mathematical. They also have homework where they get feedback and can rework it. <p> Frédéric talked about the French system, where there is a competative examination after the first year of teacher training (fourth year of university studies). Therefore, history of mathematics must also be suited to help students pass the exam. Major aims are then: - Enightening students knowledge in mathematics - Putting distance between students and mathematical knowledge - Link mathematical knowledge to other school disciplines - Training through (useful and job connected) research<p> Then: why assess, how assess, and is there any point in assessing the HM components, if the goal is not to learn HM? He gave examples of tasks he gives students and where it is quite simple to give similar tasks for the purpose of assessment - but still without being certain that it makes sense. <p> Ysette talked about maths camps where they got very few applicants, because of using the words "for gifted students". Asking students, they connected "gifted students" to the grades. These were attitudes that are problematic in teachers. History of mathematics must combat this, not strengthen it. <p> Concretely, comparing different textbooks' presentation of historical facts was an example on how to work on history of mathematics, which can be done in many different ways depending on students' background. They assess using lesson planning based on an historical excerpt.<p> Finally, David talked on experiences with preservice secondary school students, where the goal was to obtain deorientation (see his oral presentation). Assessment is problematic. Reading of original sources is a hermeneutic extreme sport without helmet. Students suffer the original sources, it includes shock and violence. Otherness is linked to empathy, keeping the subjectivity of the other alive. How to have the students suffer and maintain the empathy, how to assess the students?<p> Often, the evaluation is done by giving another text of one of the same authors they have worked on, trying to see if they understand the new text. Then the adventure ends. How to avoid that the relationship to the history ends?<p> Ewa Lakoma gave some input from her experiences in Poland, where examinations were given asking about historical facts, and group examinations where social skills were taken into account. <p> Evelyne reacted to the question of whether history of mathematics is a Humanities subject. HM is not in the tradition of mathematics, and helps us to speak mathematics and talk about mathematics with others. Mathematics is otherwise often taught by writing mathematics on the blackboard and by correcting papers with small signs on papers that have been handed in.<p> Janet pointed out that when we read mathematics, we have some techniques (for instance, finding an example) which students do not have. Therefore, many have turned to "guided reading" through tasks. (Actually, an interesting research project would be to study how different authors design the "guided reading" tasks and how they describe their reasoning between these tasks in their papers in HPM. Interviews would of course also be a possibility. It could also be compared to how this is done in other fields, such as Norwegian (L1) education.)<p> Tinne pointed out that the learning goals could both be on the big scale and on the small scale (the single session), and the assessments will vary according to these.<p> After lunch (and a much needed Starbucks ice coffee from the local minimarket), I chose the workshop of Caroline Kuhn and Snezana Lawrence on "Personalised Learning Environment and the History of Mathematics in the Learning of Mathematics". Personal is in the sense that the learner does design his own learning environment, choose tools etc.<p> After an introduction round, we had a round about our expectations. By this time, half an hour had passed. Then Caroline discussed different definitions of PLE. Ownership and agency are interesting in this context. Building a PLE is a good learning activity.<p> Four examples of websites on mathematics (which are not PLEs): Mathisgoodforyou.com NRICH MacTutor - StAndrews Mathigon.org<p> PLE outside mathematics: graasp Plenk 2010: connect.downes.ca/how.htm (Not PLE but a MOOC, which is simething different altogether.) Some other PLE: - works, but it is not so easy for students. (Valtonen Teemu)<p> After the break, in which a few of us was honestly still at a loss to understand exactly what a PLE was, we were put in groups and asked to craft an example of a teaching strategy for the tangent line problem, and then given a few sources (Apollonius and Fermat) and asked how knowledge of the sources could enhance the teaching. We all agreed that it couldn't really enhance the particular teaching ideas that we had thought of at first. This is, of course, often a problem for teachers: even though there are millions of sources online, the sources you find may not be interesting for your immediate purpose.<p> After the workshop, my thoughts are that my iPad (as well as my PC) can be regarded as my personal learning environment, with selected websites (bookmarks) and programs/apps, as well as resources I have made on my own. However, to be more useful in education, such a PLE needs an easy way for the teacher to suggest resources for the students, and - more importantly - it needs an ecosystem into which students may put their products and where other students may find these. A PLE with unlimited and uncurated access to "everything" will be fine for the expert, but certainly not for the student.<p> In the workshop, BBC's A Brief History of Mathematics was mentioned. I will have to check that out.<p> Thus ended Day 4. I skipped the oral presentations that started at 5:30, and waited for the Happy Hour instead...Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-32417997589276905972014-07-16T21:52:00.001-07:002014-07-16T21:52:37.529-07:00ESU7 Day 3The first of Wednesday's talks was Cécile de Hosson's on "Promoting interdisciplinary teaching through the use of elements of Greek and Chinese early cosmologies." Her talk discussed the connections between physics and mathematics, and how the history of physics could be used as part of physics education. She gave examples, for instance Eratosthenes' calculation of the circumference of the Earth, which is of course a good example that should be part of mathematics education. She noted that students had trouble accepting that the sun's rays are "nearly parallell" - mathematically, "nearly parallell" is quite different from "parallell"! This is of course a central part of the tension between mathematics and physics. The talk was one of several at the conference highlighting not just that the use of history will have to be different for different target groups, but also how it will have to be different.<p> Then there was the first panel; "Computational Technology: Historical and philosophical approaches to technics and technology in mathematics and mathematics education". I'm always very sceptical of panels, as they tend to disintegrate into four or five small talks that does not neccessarily connect very much to the professed theme of the panel and very rarely touch upon what the others have said. This, however, was beautifully organized, in that each panelist gave just a short talk (5 minutes?), followed by an answer by one of the other panelists before going on to the next short talk. All panelists were in active dialogue with the others and the theme of the panel.<p> Mario Sánchez Aguilar discussed how computer technology changes the way pupils and teachers work on mathematics. One way is through the use of non-traditional sources for help in working on the mathematics - nowadays, students can get so much help in different websites, for instance. For teachers, there is the possibility to enrich the instructional techniques, bringing videos into the classroom (YouTube) and by bringing the teaching into pupils' homes.<p> In response to this, Per Jönsson asked what much of mathematics needs to be internalized and how much can be outside you for you to know mathematics. He also asked if the teacher is needed at all in this system.<p> Per then went on to discuss "what is mathematics?" Computing have changed problem solving dramatically (in the research on mathematics). Mathematics should now be learned with more focus on problem solving and computing.<p> Mikkel Willum Johansen reacted by pointing out that the role computers play is as a tool for mathematicians.<p> Mirko Maracci: computers are used for educational purposes. Two ways of looking at it is that they mediate learning processes or that they embody knowledge. When they're seen as mediating, it can be either that the artifact may permit the transformation of the object and/or permit the subject's concious-raising of the object.<p> Morten reaction to Mirko: important to be critical - tool also changes and transforms mathematics. When a tool is used, it can change things in different ways. Important to study of the whole ecosystem.<p> Mikkel pointed put that mathematics is impossible without the right tools - it is a tool-driven practice. Tools have consequences, however. So even though computers are "just another tool", they have an impact on the content of mathematics. Computer-assisted proofs, mass cooperation and experimental mathematics are examples of how mathematics is changing.<p> Morten said that in teaching, computers challenge the existing mathematics education practices, for instance in something so basic as "what is a good task?" Mathematics education is squeezed between research mathematics, school traditions, the applications in specialized domains and children's everyday domains. New tools will change all four of these domains.<p> Evelyne raised the obvious question of gender (in this all-male panel). What are the gender aspects of the influences computers have? There was quite a bit of discussion on this, although mostly anecdotal in nature.<p> Another comment from the audience: we have to decide what we are going to teach - we cannot go on teaching what we have done. And we have been in this situation before through history.<p> It was nice to have a panel about computers in mathematics education without focusing all the time on the problems they bring by stealing students' attention. Of course Facebook was mentioned, but I was on Facebook at the time, so I didn't hear the complaints so well... (I had just blogged about yesterday's experiences at the conference, and noticed that a Swedish colleague who took part in HPM in Daejon but couldn't be here, thanked me for these updates...)<p> After this panel, the academic program was over, but the day had just begun. We had a bus ride to Nyhavn, a boat trip on the canals (with lunch), a guided tour of Christiania, a couple of hours on our own and finally a dinner which transformed into a party and lasted into the small hours of the night. There were no presentations during all that time, but I had more academic conversations than for the rest of the conference combined. I had interesting chats with people from France, Norway, China, Italy, Sweden, Denmark, Germany and probably many others that I have forgotten by now. The programme is usually too tight to have long conversations in the breaks, so this day was welcome also for giving us time to discuss. And of course we managed to squeeze in some non-academic discussions as well... Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-63786071459080906412014-07-16T00:51:00.002-07:002014-07-16T00:51:46.938-07:00ESU Day 2The second day of the conference started with Evelyne Barbin's plenary talk titled "Epistemologies and Theories behind History and Education: thirty years after Hans Freudenthal". Of course, Evelyne is an important figure in the HPM group, as Jan van Maanen stressed in his introduction of her. She started by referring to Hans Freudenthal, who protested the teaching of ready-made concepts instead of the development of the concepts. The phrase he coined for this, and which was mentioned throughout the talk, was "antididactical inversions".<p> Evelyne referred to two different ways of doing history; disorienting history and rational history, where the former strives to understand the facts of history, while the latter strives to give "big" theories. The former is looking at differences between authors, the latter concentrating more on similarities. I understood Evelyne's talk as a strong criticism of many attempts at the latter, where broad generalizations on the development of mathematics are made without sufficient regard for the details of the actual development. <p> Most of her talk focussed on particular, important examples from the history of mathematics (finding of tangents of curves, Leibniz' definitions of functions, conceptions of tangents etc) and examples of researchers giving statements of the development of mathematics which do not fit with the particular details. It would be dangerous for me to try to repeat concrete examples, as I do not have Evelyne's knowledge of the details.<p> Evelyne ended by warning against the "danger of a terrible didactical inversion: the didactical inversion of history of mathematics".<p> For the day's two hour workshop, I chose to listen to Jan van Maanen. It is more than fifteen years since I heard him the first time, and I always grab the opportunity when I have the chance. His title today was "'Telling mathematics' revisited". This was a development of a talk he held fifteen years ago (in Louvain), and two of the people present had also heard the talk then. <p> As a teacher, Jan spent time on problems that the pupils brought, and after a while, he realized that they had a lot of history in them. These stories have travelled miles and years.<p> For instance, he told us that he had talked to a Dutch girl, who had been told a problem by a girl while waiting for a ski lift in AustriaAustria 2010: "If I give you 2 sheep, you will have twice as many sheep as I have. If you give me two of your sheep, we have the same number of sheep. How many do we have?" The same story can be found, in variants, for instance in Nine chapters, 179 AD, and was included in Euler's Elements of Algebra 1770.<p> He gave lots of examples of such problems, such as the broken bamboo, the hundred birds problem, the two brothers (one of which always lies and the other who always tells the truth), "as I was going to St. Ives", 3-4-5 triangles in Groningen doors (Frank Swetz p 119).<p> Why does "telling mathematics" make sense also within the school? <ul><li>Because of the challenge - not neccessarily using exactly what they learned that lesson. <li>It combines the usual with the unexpected. <li>It is the student who decides! </ul> Presently, Jan has had two workshops with mathematics teachers to let them exchange mathematics that they knew from being told, asking them to make notes, then analyzing these notes in order to sort and cathegorize the entries, and to trace the entries back to their source.<p> He did the same with us, and "my" group (Mustafa, David and me) came up with three examples: - weights (one lighter than the others) - area with given perimeter - three people with black or white star on their foreheads. The winner is the person who knows his own color. They have to clap when they see a white star. When person A open his eyes, he sees two white stars. All three persons clap. But noone claims to know their own color. Then A knows. Why?<p> There were discussion at the end about how to "use" these problems. If the teacher gives the problem, should they be part of a structured plan of teaching? Jan's main point is perhaps to have the students contribute to the content of the classes, which means they can not necessarily fit nicely into the overall plan of the teacher. The point of giving students a chance to participate is more important.<p> Personally, I'm a bit depressed that I - from my memory - almost didn't work on anything outside the textbooks for my twelve years in school.<p> One sidenote: Jan mentioned a mathematical walk of Groningen - maybe this would also be a good idea for Oslo?<p> The second workshop of the day, this time a three-hour one, was Desiree Krüger and Sara Confalonieri on German and French textbooks' handling of negative numbers. They gave a background of the "elementary" textbooks that they wanted us to work on. The textbooks in question were written for universities, but it could not be assumed that the students had any prior knowledge of mathematics. In France, though, they had military schools that included mathematical teaching.<p> Then we worked in groups on selected original sources in the form of textbooks. My group (Andreas and me) worked on Kässtner's "Anfangsgrunde der Arithmetik Geometrie ebenen und sphärischen Trigonometrie, und Perspectiv". For me, both the typeface (fraktur) and language (German) was difficult, so there were some layers of difficulties to get through in addition to the unfamiliar definitions and examples (and context, of course). Particularly interesting was that the textbook stressed that you can choose for yourself what should be considered positive and what should be considered negative whenever you have two opposing magnitudes. For instance, a debt is negative if you consider wealth as the positive, but is positive if you choose to see the debt as positive. We wondered if this way of viewing it may be better than the normal way of doing it in Norwegian textbooks today.<p> Then on to oral presentations. First, David Guillemette talked on "Sociocultural approaches in mathematics education for the investigation of the potential of history of mathematics with pre-service secondary school teachers." He takes disorientation (dépaysement) (Barbin) as a starting point. He sees a gap between theoretical and empirical research in HPM. David wants to get deeper into the concept of disorientation and see how it turns up in real situations. He will use the theory of objectivation. (Learning maths is not just learning to do maths, but to be in mathematics (Radford)). He wants to give the pre-service teachers a voice. <p> He will also discuss whether the disorientation is an individual experience or could/should be seen in a social context. The narratives will be collected into a polyphonic narration, trying to integrate different points of view. One example of a preliminary results: a feeling of otherness and empathy towards the authors and the pupils are present.<p> Afterwards, there was a short discussion on the best translation of dépaysement (or whether it should be left untranslated). Alienation has also been used.<p> Secondly, René Guitart talked on "History in Mathematics According to André Weil." Weil used original sources to explain today's comprehension, reading the original sources with the help of today's knowledge, but without pretending that those authors had the same knowledge. Weil suggested historical notes in the papers of the Bourbaki group. A historical problem is (only) considered important if it generates a method or general theory. In two letters to his sister, he explained the meaning of his mathematical work by means of its history.<p> Thirdly, Leo Rogers' theme was "Historical Epistemology: Contexts for contemplating classroom activites." His talk concerned the connection between the origin of mathematics in ancient times and the connection to teaching of kids today, with lots of theories involved and lots of practical examples of what mathematics ancient cultures were doing. Sadly, my notes are not sufficient to give a detailed summary, so it's best to wait for Rogers' article (as is of course the case of most of the talks and workshops I mention).<p> One good quote from Neugebauer which I will keep in mind, was about how we do not gain a "historical perspective" as time goes by, we are just less burdened by evidence, which means we make bolder generalizations without fear that historical sources will contradict us. With this quote, Rogers' talk ended the day on the same note as the day started.<p> After this, the Advisory Board of the HPM had its traditional dinner, which included great food and wine and many friendly conversations - as well as a little official business. Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-51219557701348482552014-07-14T21:31:00.001-07:002014-07-14T21:38:22.178-07:00ESU7: Day 1The 7th European Summer University on the History and Epistemology of Mathematics takes place in Copenhagen this week. This is my 4th ESU, so it's already a tradition for me, and it is always nice to come back to meet many well-known faces - as well as many new ones. As usual, I will try to blog from the summer university, but as usual I will have to warn you that my impressions of what people were trying to get across, may be mistaken.<p> Conference organizer Uffe Jankvist gave a short introduction to the buildings and Evelyne Barbin an introduction to the history of ESUs. This is a special occation, as it is the last ESU Evelyne will take part in before she retires. Tinne Hoff Kjeldsen then gave an introduction to the themes of this conference.<p> The first plenary speaker was Adriano Demattè, with the title "History in the classroom: educational opportunities and open questions". I know him mostly from these conferences and his teaching materials, which are very inspiring. He started by referring to Schön, and claimed that teachers have to look for proper circumstances in order to reflect. In his talk, he went on to show how history of mathematics will help them.<p> Interestingly, he spent quite some time discussing his context, how many classes he teaches and so on. This is necessary, because contexts are so different in different countries. Even though he teaches pupils aged 14-19, the topics he teaches corresponds to what Norwegian students learn at age 16-20, for instance. But also, the fact that he teaches maybe five different classes at four different levels means that his teaching will be different from a professor who has one or two courses in a semester. It is not easy to be innovative all the time in such a context. And Demattè is not content with being innovative himself, he also want other teachers to be able to teach using historical materials.<p> Demattè discussed the genetic approach vs the hermeneutic approach. He referred to Jahnke's 2014 article on how history of mathematics can be used after students have had a first meeting with a new topic. He looked at how some of the steps in Jahnke 2014 would be difficult in a classroom.<p> He then described some teaching he had done with a short part of Pacioli's Summa (1494). He asked students to read and interpret. The problem was "Find for me a number that, if added to its square, makes twelve." Pacioli solved this by an algorithm. Of course, he ends up with square root of 12 1/4 minus 1/2, thus three. But there were no modern symbols, and the modern division into question and solution were not as obvious. <p> Not all students thought about reading the text several times to understand more, and many did not compare with the modern solution to try to understand words (such as "root"). There was no hermeneutic circle - students did not make guesses, and did not move between parts and the whole of the document. It seemed that only those students who was confidence that their answer would be accepted byt he rest of the class, had the willingness to guess.<p> Demattè's proposed compromise: a textbook where every chapter contains a "historical laboratory", with a list of prepared questions to guide the students in their interpretation, making sure they move between parts and the whole, work on central words in the text and so on. The questions are aimed at facilitating students' work, take into account typical student prerequisites, bridge between modern solution and Pacioli's solution. He also mentioned that these questions could be used for tests, which is really interesting - and controversial, I would think.<p> A comment from me (which I didn't raise there): A question is of course whether adopting the question-answer format of a traditional textbook helps students or just turns this into just another hunt for the right answers. But again, this is a case where such a compromise may be needed to make it feasable to use it in ordinary Italian classrooms. In Norway at least, we could add the fact that teachers might not fare so much better than the students in interpreting original sources.<p> I'm sure many would argue that the struggle with the text is an important part of students' work, and that removing the struggle also removes much of the point of working on original sources. Renaud Chorlay asked whether students improve in their reading/interpretation capabilities over time, but Adriano Dematté could not answer as he has changed his approach over time, so he does not have long-term experiences in this. Probably, what Chorlay was hinting at was precisely that the initial "suffering" of the students in working on a text without being shown the way, is worth it in the long(er) run.<p> So already there were big, open questions going through my mind...<p> The first workshop I attended was my own. I was very happy to have my workshop this early, as it is always a relief to have finished. The goal of the workshop was to have people familiarize themself with the framework on Mathematical Knowledge for Teaching (MKT), see several examples of how history can be used in mathematics education, and discuss different reasons for doing that - in light of MKT. I was quite happy with how the discussions unfolded through the two hours, although some participants were at times at a loss to understand what my aim was - as I didn't have any particular opinion to "sell".<p> One important discussion is how the HPM community should interact with the mathematics education community as a whole, for instance regarding MKT. Is MKT "bullshit" (reducing mathematics to solving problems and disregarding emotions, for instance), as someone said during group discussions, or is it an a useful framework also for us in discussing HPM? And is it important enough in the maths education world that we should point out the place of history of mathematics in it even if we're not fans?<p> One outcome of the discussions was to see how a single source could be viewed as contributing to knowledge in many different domains, depending on your context and your point of view. Several participants found MKT useful as a tool for discussing HPM outcomes in teacher education, as long as we don't see the domains as discrete and separate. The idea of using MKT for testing teachers' MKT, however, (as is one branch of the MKT research) met more scepticism.<p> Then there was a three-hour workshop by Susanne Spies. Gregor Nickel and Henrike Allmendinger from Germany: "Using original sources in teachers education - An analysis on possible effects and experiences." In a way, this workshop was a bit similar to my own, in that it also focussed on concrete examples of using history and on discussing possible outcomes. They started with the point that school education of mathematics is - and have almost always been - insufficient for starting on higher education in mathematics. In particular reflection and the ability to judge about mathematics is lacking in students, and these are fields in which history could be helpful.<p> They gave several ways in which historical sources could be a tool for teacher education (telling anecdotes (entertain, comfort) - can become heroic or jovial; genetic use (Toeplitz): implicit or explicit - can be confusing;defamiliarizing; paradigmatic use) and how it can be a discipline for reflection (historico-critical perspective; history of ideas and culture perspective (secondary sources?))<p> They listed many examples from high school teachers' education where they included historical sources (real analysis; lecture on different solutions of the isoperimetric problem; exercises on irrational numbers and incommensurability; worksheet on Bernoulli's way of finding the tangent of a parabola. Reflecting on different ways of argumentation; worksheet on different interpretations of the derivative; presentations of students based on historical sources: training text comprehension of original sources.<p> They use historical sources in courses on elementary mathematics, advanced mathematics, history of mathematics and pedagogy of mathematical, and were particularly interested in whether historical sources had different roles to play in the different kinds of courses.<p> Then we worked in groups on a part of Euclid's Elements: theorem 47 (Pythagoras' theorem). We worked in groups discussing which contexts the source could be used in and what impacts it could have. As someone commented also in my workshop, looking at single sources and their impact risks forgetting the more overarching impact of using history of mathematics repeatedly. For instance, will students really change their view of the epistemology of mathematics based on one example from history of mathematic?. Probably not - it is the 13th example that makes the difference (I'm joking about the number - I think...). Thus, when looking at single sources, we look at the particular impact of the source, and not on the general impact of HM.<p> Thus, we focused on what this particular part of Euclid could contribute, and all groups agreed (it seemed to me) that a likely output could be on understanding the concept of proof (through time) and on being able to interpret different texts (which a teacher needs to be able to do when his students do surprising things).<p> Then: the use of philosophical texts in maths education. We did a similar exercise with a work of Plato (seventh letter) on knowledge of objects (name, definition, image, knowledge - and the fifth). We discussed how this text could be used in different courses. Of course, it could inspire discussion about the insufficiency of just giving a definition to students, but an important question is why this discussion would be better using the historical source than without? (In a way, I wondered, is it the case that in research mathematics, a concept IS its definition to a higher degree than in school mathematics?)<p> The evening's oral presentation was chaired by me, and I seemed not to be able to chair and make notes at the same time, so my comments on those two presentations are a bit short. Panagiotis Delikanlis talked about how he works on the mathematical problem contained in Plato's Meno, while Fanglin Tian talked about how they work in a lesson study-like way on teaching logarithms based on the origin of logarithms. Although both were based on history, the second was much more connected to the historical context. I'm looking forward to reading her article with (probably) some more detail.<p> Then there were two poster presentations. The first, by Paulo Davidson, connected the themes HM, Funds of Knowledge and Culturally Relevant Pedagogy. It would be useful if this presentation was longer to get more details, for instance on how history was used implicitly to work on algebra in new ways. Moreover, I should read more about CRP, for instance an article cited by Ladson-Billings. The second, by Julio Corrêa, was on mathematics, education, modernism and war. In the short time he had, he could just sketch some of the connections between these concepts. The "Math Wars" in the US is an example, which is a debate (not really a "war", but with war-like rhetoric ("a nation at risk").)<p> Thus ended the scientific programme of the first day. But then there was Happy Hour, and I was happy to talk to many friendly and interesting people. It was particularly nice to talk to Torkel Heiede on his home turf - he noted that he'd worked in these buildings for thirty years. In the opening of the summer university, it was mentioned that the 2016 HPM conference will be held in Montpellier in France, so that will be my next chance of meeting many of these people.<p> I'll continue blogging tomorrow... <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0tag:blogger.com,1999:blog-2210497064207378386.post-58305083021897529932013-08-02T00:12:00.001-07:002013-08-02T00:12:03.663-07:00PME37 Day 5 #pme37Tim Rowland's talk on "Developing one-to-one teacher-student interaction in post-16 mathematics research" started the fifth day of PME37. He talked about the work of Clarissa Grandi. Her research was motivated by a feeling of mismatch between her intentions and her practice, where she felt she helped the students "too much", instead of scaffoldig. Rowland noted that "teaching by telling" is very prevalent, has been widely critizised but that there are not necessarily easily available and easily implementable alternatives for a teacher who wants to do something else.<p> Grandi analysed the form and function of her utterances during one-to-one interviews with students (and she had a few other data sources as well). It is an interesting choice to do this in the form of interviews instead of "real" classroom situations. I can see how the analysis of her utterances in interviews can give insights that are transferable to her work in the classroom. In the end, she had 12 codes showing different functions. <p> In the first cycle, there was a preponderance of "telling". Grandi was not happy with this. She used "telling" to demonstrate, direct, explain and funnel, not only to confirm, discuss conventions or "parallell model" (involving the solution of a related, but often simpler, example), in which cases telling was more meaningful, according to her. In the second cycle, she demonstrated and explained considerably less.<p> Through the research, Grandi realized that telling is sometimes consistent with a social constructivist agenda, but that she had also been able to modify her behaviour in the direction she wanted to.<p> Of course, some of the discussion afterwards was on the problem of doing research on oneself, and the problems you then have of distancing yourself from the situations.<p> In short oral presentations afterwards, first Jajaluxmi Naidoo presented a study on pre-service teachers' understanding of effective teaching strategies. The study showed that students with no teaching experience thought they should use the same strategies as their teachers had used in school. They did not have much belief in the more learner-centered strategies - it wouldn't work in real life. Which may be correct, South African classes often have more than 90 pupils, often with more ages in the same class and with many of the children hungry and unconcentrated. Of course, this is also relevant for other countries, although not as extreme- many students regard the strategies that teacher education advocates as unrealistic in that they do not take into account all the difficulties present in a classroom. This is a significant challenge for teacher education. We cannot include all the complexities of the classroom all the time in our discussions, but neither should we disregard them completely.<p> Charlotte Krog Skott gave a talk on "Lesson study in teacher education". This is based on an attempt to adopt the Japanese practice of lesson study to teacher education, where teacher students experiment with and discuss teaching. In analysis, they used cathegories from the knowledge quartet and others, and also used elements from the theories of communities of practice and situated learning. The pre-service students guided the students in a certain direction, in spite of their intentions. It also became obvious that the pre-service teachers didn't discuss the mathematics in the lesson study, instead focusing on how they could have asked questions to have the students think instead. They also gave a good example where PSTs designed a lesson where students should rotate to see four different tasks, and the PSTs were very good at picking different representations and making varied activities, that sadly did not connect to each other mathematically. Another result was that there is a potential to develop a community of practice around lesson study.<p> In just a few weeks' time, there is the ECER conference in Istanbul. As we would have to leave Kiel Friday morning, we spent the rest of Thursday planning the presentation in Istanbul. We had a very productive meeting at the hotel balcony, and now feel (almost) ready for ECER...<p> It was nice to be at PME - it was a lot friendlier than I thought it would be, and Germany is a very nice country (I'm already looking forward to ICME in 2016). I will always remember this conference for my struggles with Norwegian authorities during the conference, I'm afraid, but the conference as such was quite good.<p>Bjørnhttp://www.blogger.com/profile/16158361595130866728noreply@blogger.com0